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

<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="scmp_gcd.html#K2">GBP</a> ;<br/>
<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6">INT</a> ;<br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="ami_1.html#NM4">State</a> of <a href="scmpds_2.html#K1">SCMPDS</a> ;<br/><b>given </b><font color="Maroon">c<sub>17</sub></font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6">INT</a> <b> such that </b><a NAME="E1:9_1"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:9_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><b>assume </b><a NAME="E3:9_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/><b>thus </b><a NAME="E4:9_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <a href="scmp_gcd.html#K2">GBP</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scpinvar.html#E2:9_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scpinvar.html#E10"><i><font color="Green">Lemma6</font></i></a></i>;<br/><b>thus </b><a NAME="E5:9_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p3">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="scmpds_6.html#T34">SCMPDS_6:34</a>, <a class="ref" href="scmpds_6.html#T35">SCMPDS_6:35</a></i>;<br/><b>thus </b><a NAME="E6:9_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scpinvar.html#E2:9_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scpinvar.html#E10"><i><font color="Green">Lemma6</font></i></a></i>;<br/><b>thus </b><a NAME="E7:9_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6">INT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6">INT</a> <b> such that </b><br/><a NAME="E2:9_1_1"/><i><font color="Green">E13</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>18</sub></font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="scpisort.html#T3">SCPISORT:3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>thus </b><a NAME="E3:9_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="scpisort.html#R1">is_FinSequence_on</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:9_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p3">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scpinvar.html#E2:9_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scpinvar.html#E10"><i><font color="Green">Lemma6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/>

<b>set </b><font color="Maroon">c<sub>19</sub></font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="wsierp_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<div><i><font color="Green">E14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:9_1_1_2"/><i><font color="Green">E15</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_2_1"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_2_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_2_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_2_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:9_1_1_2_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E4:9_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:9_1_1_2_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>18</sub></font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="scpinvar.html#E1:9_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scpinvar.html#E1:9_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="scpinvar.html#E3:9_1_1_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="scpinvar.html#E5:9_1_1_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:9_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E9:9_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>20</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="scpinvar.html#E7:9_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E10:9_1_1"/><i><font color="Green">E16</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E11:9_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font> <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="E12:9_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E13:9_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="scpinvar.html#E11:9_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<font color="Maroon">c<sub>18</sub></font> <a href="wsierp_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
<font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="scpinvar.html#E10:9_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="scpisort.html#D1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E3:9_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="scpinvar.html#E2:9_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="scpinvar.html#E13:9_1_1"><i><font color="Green">E18</font></i></a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p2">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="wsierp_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scpinvar.html#E2:9_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scpinvar.html#E10"><i><font color="Green">Lemma6</font></i></a></i>
<br/>.= 
 <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scpinvar.html#E9:9_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
;<br/><br/>


</div><b>end;</b></div><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:9_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E2:9_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:9"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p5">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span>,<font color="Maroon">c<sub>16</sub></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">c<sub>18</sub></font></span>)</span> = 
<font color="Maroon">c<sub>16</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="scpinvar.html#E2:9_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="scpinvar.html#E10"><i><font color="Green">Lemma6</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="scpinvar.html#E3:9_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="scpinvar.html#E1:9_1_2"><i><font color="Green">E13</font></i></a></i>
;<br/><br/></div>
<b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9">IExec</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K2">while&lt;0</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p0">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p5">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p4">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="scmpds_2.html#K2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="scmpds_8.html#K2">while&lt;0</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span> <a href="scmpds_6.html#R1">is_closed_on</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="scmpds_8.html#K2">while&lt;0</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_2.html#K13">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,1,<span class="p4">(<span class="default"><a href="scmp_gcd.html#K1">intpos</a> 3</span>)</span>,0</span>)</span> <a href="scmpds_4.html#K6">';'</a> <span class="p3">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,2,1</span>)</span></span>)</span> <a href="scmpds_4.html#K5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K12">AddTo</a> <a href="scmp_gcd.html#K2">GBP</a> ,3,1</span>)</span></span>)</span> <a href="scmpds_6.html#R2">is_halting_on</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="scpinvar.html#E1:9"><i><font color="Green">E9</font></i></a>, <a class="txt" href="scpinvar.html#E11"><i><font color="Green">Lemma7</font></i></a>, <a class="ref" href="scpinvar.html#T8" target="_self">Th8</a></i>;<br/>


</div>
