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<b>let </b><font color="Maroon" title="c3">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon" title="c4">I</font> be  <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a> <a href="scmpds_5.html#V3" title="SCMPDS_5:attr.3">No-StopCode</a> <a href="scmpds_4.html#NM1" title="SCMPDS_4:NM.1">Program-block</a>;<br/><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> ;<br/><b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">c</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/><b>let </b><font color="Maroon" title="c10">m</font>, <font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c12">m1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/>











<b>set </b><font color="Maroon" title="c13">b</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font>;<br/>
<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E16">A1</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> )
 ;<br/>

<b>assume </b><a NAME="E2:16"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c10">m</font>
 ;<br/>

<b>assume </b><a NAME="E3:16"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c10">m</font>
 ;<br/>

<b>assume </b><a NAME="E4:16"/><i><font color="Green" title="E19">A4</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E5:16"/><i><font color="Green" title="E20">A5</font></i>: 
( <font color="Maroon" title="c12">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c12">m1</font></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E6:16"/><i><font color="Green" title="E21">A6</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <br/> for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <br/> for <font color="Olive" title="b4">k1</font>, <font color="Olive" title="b5">k2</font>, <font color="Olive" title="b6">y1</font>, <font color="Olive" title="b7">yn</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b4">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Olive" title="b5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b4">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b5">k2</font></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b7">yn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">y1</font> &amp; <font color="Olive" title="b7">yn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) or <font color="Olive" title="b6">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b7">yn</font> ) holds <br/>( <font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> &amp; ( for <font color="Olive" title="b8">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b8">j</font> &amp; <font color="Olive" title="b8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b4">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b6">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b7">yn</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b4">k1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; ( for <font color="Olive" title="b8">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b8">j</font> &amp; <font color="Olive" title="b8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span></span>)</span> ) ) ) &amp; ( <font color="Olive" title="b6">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">yn</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b4">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 &amp; ( for <font color="Olive" title="b8">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b8">j</font> &amp; <font color="Olive" title="b8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">y1</font> ) or ( <font color="Olive" title="b7">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b8">j</font> &amp; <font color="Olive" title="b8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">j</font></span>)</span></span>)</span> ) &amp;  ex <font color="Olive" title="b8">ym</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( <font color="Olive" title="b6">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b8">ym</font> &amp; <font color="Olive" title="b8">ym</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b7">yn</font> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b5">k2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">ym</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">ym</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b7">yn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b9">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b6">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b9">j</font> &amp; <font color="Olive" title="b9">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b8">ym</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">ym</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b9">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b8">ym</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b9">j</font> &amp; <font color="Olive" title="b9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b7">yn</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b8">ym</font></span>)</span></span>)</span> ) ) ) ) &amp; ( <font color="Olive" title="b2">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c10">m</font> &amp; <font color="Olive" title="b3">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Olive" title="b1">t</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> implies <font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c14">WH</font> =  <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font>;<br/>
<b>set </b><font color="Maroon" title="c15">sWH</font> =  <a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c16">iWH</font> =  <a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <br/> for <font color="Olive" title="b2">f1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <br/> for <font color="Olive" title="b3">k1</font>, <font color="Olive" title="b4">k2</font>, <font color="Olive" title="b5">y1</font>, <font color="Olive" title="b6">yn</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b3">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Olive" title="b4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b3">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">y1</font> &amp; <font color="Olive" title="b6">yn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) or <font color="Olive" title="b5">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b6">yn</font> ) &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Olive" title="b4">k2</font></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b6">yn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b6">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; <font color="Olive" title="b2">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> holds <br/> ex <font color="Olive" title="b7">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b8">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b7">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b7">k</font> &amp; <font color="Olive" title="b8">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b7">k</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Olive" title="b2">f1</font>,<font color="Olive" title="b8">f2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b8">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> <font color="Olive" title="b5">y1</font>,<font color="Olive" title="b6">yn</font> &amp; ( for <font color="Olive" title="b9">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b5">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b9">j</font> &amp; <font color="Olive" title="b9">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">y1</font> ) or ( <font color="Olive" title="b6">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b9">j</font> &amp; <font color="Olive" title="b9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><font color="Olive" title="b8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b9">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b9">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b5">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b6">yn</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b9">j</font> &amp; <font color="Olive" title="b9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Olive" title="b8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b9">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b9">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b9">j</font> &amp; <font color="Olive" title="b9">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b3">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b7">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b9">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b9">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b7">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b7">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> );<br/>
<a NAME="E7:16"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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" title="c17">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon" title="c18">f1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/><b>let </b><font color="Maroon" title="c19">k1</font>, <font color="Maroon" title="c20">k2</font>, <font color="Maroon" title="c21">y1</font>, <font color="Maroon" title="c22">yn</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E23">A8</font></i>: 
( <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon" title="c20">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">y1</font> &amp; <font color="Maroon" title="c22">yn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) or <font color="Maroon" title="c21">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">yn</font> ) &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c20">k2</font></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">yn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c22">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c17">t</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> )
 ;<br/>

<a NAME="E2:16_1"/><b>then </b><i><font color="Green" title="E24">A9</font></i>: 
( <font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/>
<a NAME="E3:16_1"/><i><font color="Green" title="E25">A10</font></i>: 
( <font color="Maroon" title="c21">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">yn</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">k1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) ) )
 
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c23">k</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2;<br/>

<b>set </b><font color="Maroon" title="c24">tk</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">k</font>;<br/>
<a NAME="E4:16_1"/><i><font color="Green" title="E26">A11</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">k</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 
<b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="scpqsort.html#T2" target="_self" title="SCPQSORT:th.2">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">f2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E6:16_1"/><i><font color="Green" title="E27">A12</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f2</font> holds <br/><font color="Maroon" title="c25">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="scpisort.html#T2" title="SCPISORT:th.2">SCPISORT:2</a></i>;<br/>
<a NAME="E7:16_1"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c25">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c17">t</font>,<font color="Maroon" title="c10">m</font>
 
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c25">f2</font>
;<br/>

<b>thus </b><a NAME="E8:16_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">k</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="scpqsort.html#T2" target="_self" title="SCPQSORT:th.2">Th2</a></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" title="c26">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">i</font> &amp; <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f2</font> )
 ;<br/><b>hence </b><font color="Maroon" title="c25">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16_1"><i><font color="Green" title="E27">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E4:16_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:16_1"/>
<font color="Maroon" title="c25">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">k</font>,<font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>;<br/>

<b>thus </b><a NAME="E11:16_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E6:16_1"><i><font color="Green" title="E27">A12</font></i></a></i>;<br/>

<b>thus </b><a NAME="E12:16_1"/>
<font color="Maroon" title="c18">f1</font>,<font color="Maroon" title="c25">f2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="scpqsort.html#E6:16_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_1"><i><font color="Green" title="E28">A13</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:16_1"/>
<font color="Maroon" title="c25">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c21">y1</font>,<font color="Maroon" title="c22">yn</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="scpisort.html#T1" title="SCPISORT:th.1">SCPISORT:1</a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>

<b>thus </b><a NAME="E14:16_1"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c21">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">y1</font> ) or ( <font color="Maroon" title="c22">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><font color="Maroon" title="c25">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c26">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_1_2"/><i><font color="Green" title="E29">A14</font></i>: 
( <font color="Maroon" title="c21">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">yn</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font> &amp; <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> )
 ;<br/><b>hence </b><font color="Maroon" title="c25">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">j</font> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16_1"><i><font color="Green" title="E27">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_1_2"><i><font color="Green" title="E29">A14</font></i></a></i>
;<br/><br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c26">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_1_3"/><i><font color="Green" title="E29">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font> &amp; <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c17">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E4:16_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>
<br/>.= 
<font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_1_3"><i><font color="Green" title="E29">A15</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E17:16_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="scpqsort.html#E4:16_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>

<b>thus </b><a NAME="E18:16_1"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c23">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="scpqsort.html#E4:16_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E23">A16</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" title="c17">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2"/><i><font color="Green" title="E24">A17</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">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" title="c18">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon" title="c19">f1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/><b>let </b><font color="Maroon" title="c20">k1</font>, <font color="Maroon" title="c21">k2</font>, <font color="Maroon" title="c22">y1</font>, <font color="Maroon" title="c23">ym</font>, <font color="Maroon" title="c24">yn</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1"/><i><font color="Green" title="E25">A18</font></i>: 
( <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon" title="c21">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">y1</font> &amp; <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) or <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">yn</font> ) &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c21">k2</font></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">yn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c24">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c19">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c18">t</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">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" title="c24">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c24">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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" title="c24">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_1"/>
 ex <font color="Olive" title="b1">kk</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">kk</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">kk</font> &amp; <font color="Olive" title="b2">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">kk</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c19">f1</font>,<font color="Olive" title="b2">f2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c22">y1</font>,<font color="Maroon" title="c24">yn</font> &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">yn</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">kk</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">kk</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">kk</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</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"><b>suppose </b></a><a NAME="E1:16_2_1_1_2"/><i><font color="Green" title="E26">A19</font></i>: 
<font color="Maroon" title="c24">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E27">A20</font></i>: 
<font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">y1</font>
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><a NAME="E3:16_2_1_1_2"/><i><font color="Green" title="E28">A21</font></i>: 
( <font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c18">t</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c25">It</font> =  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font>;<br/><a NAME="E4:16_2_1_1_2"/><i><font color="Green" title="E29">A22</font></i>: 
( <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp;  ex <font color="Olive" title="b1">ym</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">ym</font> &amp; <font color="Olive" title="b1">ym</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c21">k2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">ym</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">ym</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">yn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">ym</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">ym</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">ym</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">ym</font></span>)</span></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c26">m1</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2;<br/><b>set </b><font color="Maroon" title="c27">t1</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>;<br/><a NAME="E5:16_2_1_1_2"/><i><font color="Green" title="E30">A23</font></i>: 
( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="ref" href="scpqsort.html#T2" target="_self" title="SCPQSORT:th.2">Th2</a></i>;<br/><b>consider </b><font color="Maroon" title="c28">ym</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:16_2_1_1_2"/><i><font color="Green" title="E31">A24</font></i>: 
( <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ym</font> &amp; <font color="Maroon" title="c28">ym</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c21">k2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">ym</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">ym</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">yn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">ym</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c28">ym</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">ym</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E4:16_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><b>set </b><font color="Maroon" title="c29">k3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><b>set </b><font color="Maroon" title="c30">yd</font> = <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E8:16_2_1_1_2"/><i><font color="Green" title="E32">A25</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>;<br/><a NAME="E9:16_2_1_1_2"/><i><font color="Green" title="E33">A26</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E4:16_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>;<br/><a NAME="E10:16_2_1_1_2"/><i><font color="Green" title="E34">A27</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E4:16_2_1_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>;<br/><a NAME="E11:16_2_1_1_2"/><i><font color="Green" title="E35">A28</font></i>: 
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c28">ym</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E12:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E36">A29</font></i>: 
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">y1</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E37">A30</font></i>: 
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:16_2_1_1_2"/><i><font color="Green" title="E38">A31</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>;<br/><a NAME="E15:16_2_1_1_2"/><i><font color="Green" title="E39">A32</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">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" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">yn</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>;<br/><a NAME="E16:16_2_1_1_2"/><i><font color="Green" title="E40">A33</font></i>: 
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E17:16_2_1_1_2"/>
<font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E18:16_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E16:16_2_1_1_2"><i><font color="Green" title="E40">A33</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E19:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E41">A34</font></i>: 
<font color="Maroon" title="c24">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><b>consider </b><font color="Maroon" title="c31">f2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E21:16_2_1_1_2"/><i><font color="Green" title="E42">A35</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">f2</font> holds <br/><font color="Maroon" title="c31">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="scpisort.html#T2" title="SCPISORT:th.2">SCPISORT:2</a></i>;<br/><a NAME="E22:16_2_1_1_2"/>
<font color="Maroon" title="c31">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E21:16_2_1_1_2"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c32">kl</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> , <font color="Maroon" title="c33">f3</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E24:16_2_1_1_2"/><i><font color="Green" title="E43">A36</font></i>: 
( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font> &amp; <font color="Maroon" title="c33">f3</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c31">f2</font>,<font color="Maroon" title="c33">f3</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c33">f3</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c24">yn</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">yn</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2"><i><font color="Green" title="E24">A17</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E8:16_2_1_1_2"><i><font color="Green" title="E32">A25</font></i></a>, <a class="txt" href="scpqsort.html#E10:16_2_1_1_2"><i><font color="Green" title="E34">A27</font></i></a>, <a class="txt" href="scpqsort.html#E13:16_2_1_1_2"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="scpqsort.html#E14:16_2_1_1_2"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="scpqsort.html#E15:16_2_1_1_2"><i><font color="Green" title="E39">A32</font></i></a>, <a class="txt" href="scpqsort.html#E19:16_2_1_1_2"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="scpqsort.html#E21:16_2_1_1_2"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><b>set </b><font color="Maroon" title="c34">t2</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font>;<br/><a NAME="E25:16_2_1_1_2"/><i><font color="Green" title="E44">A37</font></i>: 
<font color="Maroon" title="c28">ym</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c35">yc</font> = <font color="Maroon" title="c28">ym</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E27:16_2_1_1_2"/><i><font color="Green" title="E45">A38</font></i>: 
<font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/><a NAME="E28:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E46">A39</font></i>: 
<font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E29:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E47">A40</font></i>: 
<font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c36">jj</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E30:16_2_1_1_2"/><i><font color="Green" title="E48">A41</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E31:16_2_1_1_2"/><i><font color="Green" title="E49">A42</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E32:16_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green" title="E50">A43</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <font color="Maroon" title="c21">k2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E30:16_2_1_1_2"><i><font color="Green" title="E48">A41</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">y1</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>
;<br/><b>set </b><font color="Maroon" title="c37">jj</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2;<br/><a NAME="E34:16_2_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E35:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E51">A44</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E36:16_2_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E31:16_2_1_1_2"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green" title="E52">A45</font></i>: <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E35:16_2_1_1_2"><i><font color="Green" title="E51">A44</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">yc</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>
;<br/><a NAME="E38:16_2_1_1_2"/>
<font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E39:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E53">A46</font></i>: 
<font color="Maroon" title="c35">yc</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">y1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E40:16_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">yn</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">y1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E41:16_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c35">yc</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E39:16_2_1_1_2"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c38">km</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> , <font color="Maroon" title="c39">f4</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E43:16_2_1_1_2"/><i><font color="Green" title="E54">A47</font></i>: 
( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c38">km</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c38">km</font> &amp; <font color="Maroon" title="c39">f4</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c38">km</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c39">f4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c33">f3</font>,<font color="Maroon" title="c39">f4</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c39">f4</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c22">y1</font>,<font color="Maroon" title="c35">yc</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">yc</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><font color="Maroon" title="c39">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">yc</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Maroon" title="c39">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c38">km</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c38">km</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c38">km</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2"><i><font color="Green" title="E24">A17</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E9:16_2_1_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E29:16_2_1_1_2"><i><font color="Green" title="E47">A40</font></i></a>, <a class="txt" href="scpqsort.html#E33:16_2_1_1_2"><i><font color="Green" title="E50">A43</font></i></a>, <a class="txt" href="scpqsort.html#E37:16_2_1_1_2"><i><font color="Green" title="E52">A45</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><b>take </b><font color="Maroon" title="c40">mm</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">kl</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">km</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c41">tm</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c40">mm</font>;<br/><b>take </b><font color="Maroon" title="c42">f4</font> = <font color="Maroon" title="c39">f4</font>;<br/><i><font color="Green" title="E55">A48</font></i>:  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c40">mm</font> = 
 <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c32">kl</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">km</font></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>
<br/>.= 
 <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c38">km</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>
;<br/><b>hence </b><a NAME="E45:16_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c40">mm</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c40">mm</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a></i>;<br/><b>thus </b><a NAME="E46:16_2_1_1_2"/>
<font color="Maroon" title="c42">f4</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c40">mm</font>,<font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E44:16_2_1_1_2"><i><font color="Green" title="E55">A48</font></i></a></i>;<br/><b>thus </b><a NAME="E47:16_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">f4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a></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" title="c43">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">i</font> &amp; <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">f2</font> )
 ;<br/><b>hence </b><font color="Maroon" title="c31">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E21:16_2_1_1_2"><i><font color="Green" title="E42">A35</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E49:16_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c31">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font>,<font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>;<br/><a NAME="E50:16_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">f1</font>,<font color="Maroon" title="c31">f2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E21:16_2_1_1_2"><i><font color="Green" title="E42">A35</font></i></a></i>;<br/><a NAME="E51:16_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">f1</font>,<font color="Maroon" title="c33">f3</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="ref" href="rfinseq.html#T2" title="RFINSEQ:th.2">RFINSEQ:2</a></i>;<br/><b>hence </b><a NAME="E52:16_2_1_1_2"/>
<font color="Maroon" title="c19">f1</font>,<font color="Maroon" title="c42">f4</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="ref" href="rfinseq.html#T2" title="RFINSEQ:th.2">RFINSEQ:2</a></i>;<br/><a NAME="E53:16_2_1_1_2"/><i><font color="Green" title="E56">A49</font></i>: 
<font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E11:16_2_1_1_2"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green" title="E57">A50</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" title="c43">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_5"/><i><font color="Green" title="E58">A51</font></i>: 
( <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">j</font> &amp; <font color="Maroon" title="c43">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_5"/><b>then </b><i><font color="Green" title="E59">A52</font></i>: 
<font color="Maroon" title="c43">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_2_1_1_2_5"/><i><font color="Green" title="E60">A53</font></i>: 
<font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c43">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E53:16_2_1_1_2"><i><font color="Green" title="E56">A49</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_5"><i><font color="Green" title="E58">A51</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:16_2_1_1_2_5"/><i><font color="Green" title="E61">A54</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E13:16_2_1_1_2"><i><font color="Green" title="E37">A30</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_5"><i><font color="Green" title="E58">A51</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">yc</font> or <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">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" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_5_1_1_1"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_5"><i><font color="Green" title="E59">A52</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_5"><i><font color="Green" title="E60">A53</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" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_5_1_1_2"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_5"><i><font color="Green" title="E59">A52</font></i></a>, <a class="txt" href="scpqsort.html#E4:16_2_1_1_2_5"><i><font color="Green" title="E61">A54</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" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_5"><i><font color="Green" title="E59">A52</font></i></a>, <a class="txt" href="scpqsort.html#E4:16_2_1_1_2_5"><i><font color="Green" title="E61">A54</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">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" title="c43">i</font>, <font color="Maroon" title="c44">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_6"/><i><font color="Green" title="E58">A55</font></i>: 
( <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">i</font> &amp; <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_6"/><b>then </b><i><font color="Green" title="E59">A56</font></i>: 
<font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_2_1_1_2_6"/><i><font color="Green" title="E60">A57</font></i>: 
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2_6"><i><font color="Green" title="E58">A55</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:16_2_1_1_2_6"/><i><font color="Green" title="E61">A58</font></i>: 
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E54:16_2_1_1_2"><i><font color="Green" title="E57">A50</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_6"><i><font color="Green" title="E58">A55</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_6"><i><font color="Green" title="E59">A56</font></i></a></i>;<br/><a NAME="E5:16_2_1_1_2_6"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E54:16_2_1_1_2"><i><font color="Green" title="E57">A50</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_6"><i><font color="Green" title="E58">A55</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_6"><i><font color="Green" title="E60">A57</font></i></a></i>;<br/><b>hence </b><a NAME="E6:16_2_1_1_2_6"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_6"><i><font color="Green" title="E58">A55</font></i></a>, <a class="txt" href="scpqsort.html#E4:16_2_1_1_2_6"><i><font color="Green" title="E61">A58</font></i></a>, <a class="ref" href="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></i>;<br/></div><b>end;</b></div><a NAME="E56:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E58">A59</font></i>: 
<font color="Maroon" title="c42">f4</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c24">yn</font>
 <b>by </b><i><a class="ref" href="graph_2.html#D13" title="GRAPH_2:def.13">GRAPH_2:def 13</a></i>;<br/><a NAME="E57:16_2_1_1_2"/>
<font color="Maroon" title="c22">y1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c43">y0</font> = <font color="Maroon" title="c22">y1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E59:16_2_1_1_2"/><i><font color="Green" title="E59">A60</font></i>: 
<font color="Maroon" title="c43">y0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">yc</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><div><i><font color="Green" title="E60">A61</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" title="c44">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_7"/><i><font color="Green" title="E61">A62</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">y0</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_7"/><b>then </b>
<font color="Maroon" title="c44">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_2_1_1_2_7"/><b>then </b><i><font color="Green" title="E62">A63</font></i>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E4:16_2_1_1_2_7"/>
<font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:16_2_1_1_2_7"/><b>then </b><i><font color="Green" title="E63">A64</font></i>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E3:16_2_1_1_2_7"><i><font color="Green" title="E62">A63</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">yc</font> or <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">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" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_7"><i><font color="Green" title="E61">A62</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_7"><i><font color="Green" title="E62">A63</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" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_7"><i><font color="Green" title="E61">A62</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2_7"><i><font color="Green" title="E63">A64</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="E7:16_2_1_1_2_7"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_7"><i><font color="Green" title="E61">A62</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2_7"><i><font color="Green" title="E63">A64</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E61">A65</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" title="c44">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_8"/><i><font color="Green" title="E62">A66</font></i>: 
( <font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">f4</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_8"/><b>then </b><i><font color="Green" title="E63">A67</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">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" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">yc</font> or <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">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" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_8"><i><font color="Green" title="E62">A66</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_8_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_8_1_1_2"/>
<font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_8_1_1_2"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_8"><i><font color="Green" title="E62">A66</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_8"><i><font color="Green" title="E63">A67</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="E4:16_2_1_1_2_8"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_8"><i><font color="Green" title="E62">A66</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_8"><i><font color="Green" title="E63">A67</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E62:16_2_1_1_2"/><i><font color="Green" title="E62">A68</font></i>: 
<font color="Maroon" title="c28">ym</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E63:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E63">A69</font></i>: 
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">ym</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">ym</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E27:16_2_1_1_2"><i><font color="Green" title="E45">A38</font></i></a>, <a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E61:16_2_1_1_2"><i><font color="Green" title="E61">A65</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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> or <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_9_1_1"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">ym</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">ym</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E11:16_2_1_1_2"><i><font color="Green" title="E35">A28</font></i></a>, <a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E25:16_2_1_1_2"><i><font color="Green" title="E44">A37</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_9_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_9_1_2"/>
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_9_1_2"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">ym</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">ym</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E25:16_2_1_1_2"><i><font color="Green" title="E44">A37</font></i></a>, <a class="txt" href="scpqsort.html#E62:16_2_1_1_2"><i><font color="Green" title="E62">A68</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E65:16_2_1_1_2"/><b>then </b><i><font color="Green" title="E64">A70</font></i>: 
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">ym</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">ym</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="scpqsort.html#E63:16_2_1_1_2"><i><font color="Green" title="E63">A69</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>;<br/><div><i><font color="Green" title="E65">A71</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" title="c44">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_10"/><i><font color="Green" title="E66">A72</font></i>: 
( <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_10"/><b>then </b><i><font color="Green" title="E67">A73</font></i>: 
<font color="Maroon" title="c43">y0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_2_1_1_2_10"/>
<font color="Maroon" title="c44">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ym</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2_10"><i><font color="Green" title="E66">A72</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E4:16_2_1_1_2_10"/><b>then </b>
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">yc</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c45">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:16_2_1_1_2_10"/><i><font color="Green" title="E68">A74</font></i>: 
( <font color="Maroon" title="c43">y0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">j</font> &amp; <font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">yc</font> &amp; <font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E29:16_2_1_1_2"><i><font color="Green" title="E47">A40</font></i></a>, <a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E59:16_2_1_1_2"><i><font color="Green" title="E59">A60</font></i></a>, <a class="txt" href="scpqsort.html#E60:16_2_1_1_2"><i><font color="Green" title="E60">A61</font></i></a>, <a class="txt" href="scpqsort.html#E61:16_2_1_1_2"><i><font color="Green" title="E61">A65</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_10"><i><font color="Green" title="E67">A73</font></i></a>, <a class="ref" href="scpqsort.html#T11" target="_self" title="SCPQSORT:th.11">Th11</a></i>;<br/><a NAME="E7:16_2_1_1_2_10"/><i><font color="Green" title="E69">A75</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E6:16_2_1_1_2_10"><i><font color="Green" title="E68">A74</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E8:16_2_1_1_2_10"/><i><font color="Green" title="E70">A76</font></i>: 
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E53:16_2_1_1_2"><i><font color="Green" title="E56">A49</font></i></a>, <a class="txt" href="scpqsort.html#E6:16_2_1_1_2_10"><i><font color="Green" title="E68">A74</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:16_2_1_1_2_10"/><i><font color="Green" title="E71">A77</font></i>: 
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E29:16_2_1_1_2"><i><font color="Green" title="E47">A40</font></i></a>, <a class="txt" href="scpqsort.html#E6:16_2_1_1_2_10"><i><font color="Green" title="E68">A74</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:16_2_1_1_2_10"/><i><font color="Green" title="E72">A78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">y1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E6:16_2_1_1_2_10"><i><font color="Green" title="E68">A74</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E11:16_2_1_1_2_10"/><i><font color="Green" title="E73">A79</font></i>: 
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E6:16_2_1_1_2_10"><i><font color="Green" title="E68">A74</font></i></a>, <a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> or <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_10_1_1_1"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2_10"><i><font color="Green" title="E69">A75</font></i></a>, <a class="txt" href="scpqsort.html#E8:16_2_1_1_2_10"><i><font color="Green" title="E70">A76</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_10_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_10_1_1_2"/>
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_10_1_1_2"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2_10"><i><font color="Green" title="E69">A75</font></i></a>, <a class="txt" href="scpqsort.html#E9:16_2_1_1_2_10"><i><font color="Green" title="E71">A77</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E13:16_2_1_1_2_10"/><b>then </b>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="scpqsort.html#E6:16_2_1_1_2_10"><i><font color="Green" title="E68">A74</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>;<br/><b>hence </b><a NAME="E14:16_2_1_1_2_10"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">ym</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="scpqsort.html#E65:16_2_1_1_2"><i><font color="Green" title="E64">A70</font></i></a>, <a class="txt" href="scpqsort.html#E10:16_2_1_1_2_10"><i><font color="Green" title="E72">A78</font></i></a>, <a class="txt" href="scpqsort.html#E11:16_2_1_1_2_10"><i><font color="Green" title="E73">A79</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E66">A80</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" title="c44">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_11"/><i><font color="Green" title="E67">A81</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ym</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_11"/><b>then </b><i><font color="Green" title="E68">A82</font></i>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E11:16_2_1_1_2"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_2_1_1_2_11"/><i><font color="Green" title="E69">A83</font></i>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E62:16_2_1_1_2"><i><font color="Green" title="E62">A68</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_11"><i><font color="Green" title="E67">A81</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> or <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_11_1_1_1"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_11"><i><font color="Green" title="E67">A81</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_11"><i><font color="Green" title="E68">A82</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_11_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_11_1_1_2"/>
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_11_1_1_2"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_11"><i><font color="Green" title="E67">A81</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_11"><i><font color="Green" title="E69">A83</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="E5:16_2_1_1_2_11"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E21:16_2_1_1_2"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_11"><i><font color="Green" title="E67">A81</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_11"><i><font color="Green" title="E69">A83</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E67">A84</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" title="c44">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_12"/><i><font color="Green" title="E68">A85</font></i>: 
( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f3</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_12"/><b>then </b><i><font color="Green" title="E69">A86</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">i</font>
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> or <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_12_1_1_1"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_12"><i><font color="Green" title="E68">A85</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_12_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_12_1_1_2"/>
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_12_1_1_2"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_12"><i><font color="Green" title="E68">A85</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_12"><i><font color="Green" title="E69">A86</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="E4:16_2_1_1_2_12"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E21:16_2_1_1_2"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_12"><i><font color="Green" title="E68">A85</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_12"><i><font color="Green" title="E69">A86</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" title="c44">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_13"/><i><font color="Green" title="E68">A87</font></i>: 
( <font color="Maroon" title="c28">ym</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">i</font> &amp; <font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font> )
 ;<br/><a NAME="E2:16_2_1_1_2_13"/><b>then </b><i><font color="Green" title="E69">A88</font></i>: 
<font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_2_1_1_2_13"/><i><font color="Green" title="E70">A89</font></i>: 
<font color="Maroon" title="c44">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">f4</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_13"><i><font color="Green" title="E68">A87</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>consider </b><font color="Maroon" title="c45">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:16_2_1_1_2_13"/><i><font color="Green" title="E71">A90</font></i>: 
( <font color="Maroon" title="c28">ym</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">j</font> &amp; <font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">yn</font> &amp; <font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E67:16_2_1_1_2"><i><font color="Green" title="E66">A80</font></i></a>, <a class="txt" href="scpqsort.html#E68:16_2_1_1_2"><i><font color="Green" title="E67">A84</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_13"><i><font color="Green" title="E68">A87</font></i></a>, <a class="ref" href="scpqsort.html#T11" target="_self" title="SCPQSORT:th.11">Th11</a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><a NAME="E6:16_2_1_1_2_13"/><i><font color="Green" title="E72">A91</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E25:16_2_1_1_2"><i><font color="Green" title="E44">A37</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2_13"><i><font color="Green" title="E71">A90</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:16_2_1_1_2_13"/><i><font color="Green" title="E73">A92</font></i>: 
<font color="Maroon" title="c45">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c31">f2</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E21:16_2_1_1_2"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2_13"><i><font color="Green" title="E71">A90</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font> = 
<font color="Maroon" title="c31">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">j</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E61:16_2_1_1_2"><i><font color="Green" title="E61">A65</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_13"><i><font color="Green" title="E69">A88</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_13"><i><font color="Green" title="E70">A89</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2_13"><i><font color="Green" title="E71">A90</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E21:16_2_1_1_2"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="scpqsort.html#E6:16_2_1_1_2_13"><i><font color="Green" title="E72">A91</font></i></a>, <a class="txt" href="scpqsort.html#E7:16_2_1_1_2_13"><i><font color="Green" title="E73">A92</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>
;<br/><b>hence </b><a NAME="E9:16_2_1_1_2_13"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">ym</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="scpqsort.html#E65:16_2_1_1_2"><i><font color="Green" title="E64">A70</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2_13"><i><font color="Green" title="E71">A90</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E70:16_2_1_1_2"/>
<font color="Maroon" title="c42">f4</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c22">y1</font>,<font color="Maroon" title="c24">yn</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E7:16_2_1_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E56:16_2_1_1_2"><i><font color="Green" title="E58">A59</font></i></a>, <a class="txt" href="scpqsort.html#E66:16_2_1_1_2"><i><font color="Green" title="E65">A71</font></i></a>, <a class="ref" href="scpqsort.html#T8" target="_self" title="SCPQSORT:th.8">Th8</a></i>;<br/><b>thus </b><a NAME="E71:16_2_1_1_2"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">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" title="c44">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_2_1_1_2_14"/><i><font color="Green" title="E68">A93</font></i>: 
( <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) )
 ;<br/>

<a NAME="E2:16_2_1_1_2_14"/><i><font color="Green" title="E69">A94</font></i>: 
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) )
 </span><b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2_14"><i><font color="Green" title="E68">A93</font></i></a></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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">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" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c35">yc</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) )
 <b>by </b><i><a class="txt" href="scpqsort.html#E28:16_2_1_1_2"><i><font color="Green" title="E46">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="E70">A95</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) )
 </span><b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2_14"><i><font color="Green" title="E68">A93</font></i></a></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" title="E71">A96</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> )
 ;<br/><div class="add"><a NAME="E2:16_2_1_1_2_14_2_1_1"/><b>then </b>
<font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2_14"><i><font color="Green" title="E68">A93</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_14_2_1_1"><i><font color="Green" title="E71">A96</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2_14_2_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_2_1_2"/><i><font color="Green" title="E71">A97</font></i>: 
( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> )
 ;<br/><div class="add"><a NAME="E2:16_2_1_1_2_14_2_1_2"/><b>then </b>
<font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2_14"><i><font color="Green" title="E68">A93</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_14_2_1_2"><i><font color="Green" title="E71">A97</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="E71">A98</font></i>: 
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">y1</font> ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) )
 </span><b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2_14"><i><font color="Green" title="E68">A93</font></i></a></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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) )
 <b>by </b><i><a class="txt" href="scpqsort.html#E12:16_2_1_1_2"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c24">yn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><i><font color="Green" title="E72">A99</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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font> or <font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">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" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_4_1_1"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E4:16_2_1_1_2_14"><i><font color="Green" title="E71">A98</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_14_4_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_4_1_2"/>
<font color="Maroon" title="c28">ym</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">yn</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_4_1_2"/>
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_14"><i><font color="Green" title="E70">A95</font></i></a></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" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">yc</font> or <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">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" title="c22">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_5_1_1"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_2_1_1_2_14"><i><font color="Green" title="E69">A94</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_14_5_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2_14_5_1_2"/>
<font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c35">yc</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_2_14_5_1_2"/>
<font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_14"><i><font color="Green" title="E70">A95</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font> = 
<font color="Maroon" title="c33">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_14"><i><font color="Green" title="E70">A95</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="scpqsort.html#E5:16_2_1_1_2_14"><i><font color="Green" title="E72">A99</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>
<br/>.= 
<font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_14"><i><font color="Green" title="E68">A93</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E72:16_2_1_1_2"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Maroon" title="c22">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">yn</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Maroon" title="c42">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c44">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_2_15"/><i><font color="Green" title="E68">A100</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c44">j</font> &amp; <font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E2:16_2_1_1_2_15"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c20">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E31:16_2_1_1_2"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:16_2_1_1_2_15"/><b>then </b><i><font color="Green" title="E69">A101</font></i>: 
<font color="Maroon" title="c44">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1_1_2_15"><i><font color="Green" title="E68">A100</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c40">mm</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c32">kl</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E44:16_2_1_1_2"><i><font color="Green" title="E55">A48</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_15"><i><font color="Green" title="E68">A100</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K30" title="AMI_1:func.30">LifeSpan</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_15"><i><font color="Green" title="E68">A100</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_2_1_1_2_15"><i><font color="Green" title="E69">A101</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c18">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E5:16_2_1_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="scmpds_4.html#T23" title="SCMPDS_4:th.23">SCMPDS_4:23</a></i>
<br/>.= 
<font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c44">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E1:16_2_1_1_2_15"><i><font color="Green" title="E68">A100</font></i></a></i>
;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E74:16_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c40">mm</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="scpqsort.html#E9:16_2_1_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="txt" href="scpqsort.html#E24:16_2_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E44:16_2_1_1_2"><i><font color="Green" title="E55">A48</font></i></a></i>;<br/><b>thus </b><a NAME="E75:16_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">t</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c40">mm</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E43:16_2_1_1_2"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="scpqsort.html#E44:16_2_1_1_2"><i><font color="Green" title="E55">A48</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="E3:16_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E9:16"/><i><font color="Green" title="E24">A102</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="scpqsort.html#E7:16"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="scpqsort.html#E8:16"><i><font color="Green" title="E23">A16</font></i></a>);<br/></i>
<a NAME="E10:16"/><i><font color="Green" title="E25">A103</font></i>: 
<font color="Maroon" title="c12">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="scpqsort.html#E5:16"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<a NAME="E11:16"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b2">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c8">f</font>,<font color="Olive" title="b2">f2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c11">n</font> &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">n</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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" title="c11">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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" title="c11">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b2">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c8">f</font>,<font color="Olive" title="b2">f2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c11">n</font> &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">n</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:16"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="scpqsort.html#E4:16"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="scpqsort.html#E5:16"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="scpqsort.html#E7:16"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="scpqsort.html#E10:16"><i><font color="Green" title="E25">A103</font></i></a></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" title="c11">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c17">nn</font> = <font color="Maroon" title="c11">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E3:16_3_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">nn</font>]
 <b>by </b><i><a class="txt" href="scpqsort.html#E9:16"><i><font color="Green" title="E24">A102</font></i></a></i>;<br/><b>hence </b><a NAME="E4:16_3_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b2">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c8">f</font>,<font color="Olive" title="b2">f2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c11">n</font> &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">n</font> &amp; ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> ) ) holds <br/><font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> holds <br/><font color="Olive" title="b2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:16"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="scpqsort.html#E4:16"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="scpqsort.html#E5:16"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="scpqsort.html#E10:16"><i><font color="Green" title="E25">A103</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c17">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> , <font color="Maroon" title="c18">f2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E13:16"/><i><font color="Green" title="E26">A104</font></i>: 
( <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c18">f2</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c10">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c18">f2</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c18">f2</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c11">n</font> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> )
 ;<br/>
<b>set </b><font color="Maroon" title="c19">sk</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font>;<br/>
<b>set </b><font color="Maroon" title="c20">s1</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c21">s2</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1;<br/>
<b>set </b><font color="Maroon" title="c22">i1</font> = <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c23">i2</font> =  <a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E14:16"/><i><font color="Green" title="E27">A105</font></i>: 
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="scmpds_6.html#T21" title="SCMPDS_6:th.21">SCMPDS_6:21</a></i>;<br/>
<a NAME="E15:16"/>
 <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="scmpds_4.html#K4" title="SCMPDS_4:func.4">';'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">I</font> <a href="scmpds_4.html#K5" title="SCMPDS_4:func.5">';'</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K4" title="SCMPDS_2:func.4">goto</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scpqsort.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<a NAME="E16:16"/><b>then </b><i><font color="Green" title="E28">A106</font></i>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="scmpds_6.html#T22" title="SCMPDS_6:th.22">SCMPDS_6:22</a></i>;<br/>
<i><font color="Green" title="E29">A107</font></i>:  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T14" title="AMI_1:th.14">AMI_1:14</a></i>
<br/>.= 
 <a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></i>
<br/>.= 
 <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E16:16"><i><font color="Green" title="E28">A106</font></i></a>, <a class="ref" href="ami_1.html#D18" title="AMI_1:def.18">AMI_1:def 18</a></i>
;<br/>
<a NAME="E18:16"/>
 <a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T26" title="FUNCT_4:th.26">FUNCT_4:26</a></i>;<br/>
<a NAME="E19:16"/><b>then </b><i><font color="Green" title="E30">A108</font></i>: 
 <a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 
<b>by </b><i><a class="ref" href="ami_1.html#T81" title="AMI_1:th.81">AMI_1:81</a>, <a class="ref" href="scmpds_4.html#T57" title="SCMPDS_4:th.57">SCMPDS_4:57</a></i>;<br/>
<a NAME="E20:16"/><i><font color="Green" title="E31">A109</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="scmpds_8.html#T17" title="SCMPDS_8:th.17">SCMPDS_8:17</a></i>;<br/>
<a NAME="E21:16"/><b>then </b><i><font color="Green" title="E32">A110</font></i>: 
 <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="scmpds_6.html#T25" title="SCMPDS_6:th.25">SCMPDS_6:25</a></i>;<br/>
<i><font color="Green" title="E33">A111</font></i>:  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>
<b>by </b><i><a class="ref" href="ami_1.html#D15" title="AMI_1:def.15">AMI_1:def 15</a></i>
<br/>.= 
 <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">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#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E4:16"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="scpqsort.html#E13:16"><i><font color="Green" title="E26">A104</font></i></a>, <a class="txt" href="scpqsort.html#E17:16"><i><font color="Green" title="E29">A107</font></i></a>, <a class="ref" href="scmpds_2.html#T68" title="SCMPDS_2:th.68">SCMPDS_2:68</a></i>
<br/>.= 
 <a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E14:16"><i><font color="Green" title="E27">A105</font></i></a>, <a class="ref" href="scmpds_6.html#T23" title="SCMPDS_6:th.23">SCMPDS_6:23</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_3.html#K2" title="SCMPDS_3:func.2">inspos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E19:16"><i><font color="Green" title="E30">A108</font></i></a>, <a class="txt" href="scpqsort.html#E21:16"><i><font color="Green" title="E32">A110</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
 <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> 
<b>by </b><i><a class="txt" href="scpqsort.html#E20:16"><i><font color="Green" title="E31">A109</font></i></a>, <a class="ref" href="scmpds_6.html#T25" title="SCMPDS_6:th.25">SCMPDS_6:25</a></i>
;<br/>
<a NAME="E24:16"/><b>then </b><i><font color="Green" title="E34">A112</font></i>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">CurInstr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K5" title="AMI_1:func.5">halt</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> 
 
<b>by </b><i><a class="txt" href="scpqsort.html#E22:16"><i><font color="Green" title="E33">A111</font></i></a>, <a class="ref" href="ami_1.html#D17" title="AMI_1:def.17">AMI_1:def 17</a></i>;<br/>
<a NAME="E25:16"/><i><font color="Green" title="E35">A113</font></i>: 
 <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="scpqsort.html#E13:16"><i><font color="Green" title="E26">A104</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/>
<a NAME="E26:16"/><b>then </b><i><font color="Green" title="E36">A114</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p3">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span> is <a href="ami_1.html#V8" title="AMI_1:attr.8">halting</a>
 
<b>by </b><i><a class="txt" href="scpqsort.html#E24:16"><i><font color="Green" title="E34">A112</font></i></a>, <a class="ref" href="ami_1.html#D20" title="AMI_1:def.20">AMI_1:def 20</a></i>;<br/>
<a NAME="E27:16"/><b>then </b><i><font color="Green" title="E37">A115</font></i>: 
 <a href="ami_1.html#K12" title="AMI_1:func.12">Result</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 
<b>by </b><i><a class="txt" href="scpqsort.html#E24:16"><i><font color="Green" title="E34">A112</font></i></a>, <a class="txt" href="scpqsort.html#E25:16"><i><font color="Green" title="E35">A113</font></i></a>, <a class="ref" href="ami_1.html#D22" title="AMI_1:def.22">AMI_1:def 22</a></i>;<br/>
<b>thus </b><a NAME="E28:16"/>
 <a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E26:16"><i><font color="Green" title="E36">A114</font></i></a>, <a class="ref" href="scmpds_6.html#D3" title="SCMPDS_6:def.3">SCMPDS_6:def 3</a></i>;<br/>

<a NAME="E29:16"/><i><font color="Green" title="E38">X</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c11">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="scpqsort.html#E1:16"><i><font color="Green" title="E16">A1</font></i></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" title="c24">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:16_7"/>
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g</font>
 ;<br/><a NAME="E2:16_7"/><b>then </b><i><font color="Green" title="E39">A116</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font> &amp; <font color="Maroon" title="c24">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="scpqsort.html#E29:16"><i><font color="Green" title="E38">X</font></i></a></i>;<br/><a NAME="E3:16_7"/><i><font color="Green" title="E40">A117</font></i>: 
 <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span>,<font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span>
 <b>by </b><i><a class="txt" href="scpqsort.html#E27:16"><i><font color="Green" title="E37">A115</font></i></a>, <a class="ref" href="scmpds_4.html#D8" title="SCMPDS_4:def.8">SCMPDS_4:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">a</font> = <font color="Maroon" title="c24">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon" title="c26">xi</font> =  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">a</font></span>)</span>;<br/><a NAME="E5:16_7"/><i><font color="Green" title="E41">A118</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="scmpds_6.html#T1" title="SCMPDS_6:th.1">SCMPDS_6:1</a></i>;<br/><div><i><font color="Green" title="E42">A119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_7_1"/>
 <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c27">l</font> =  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">a</font></span>)</span> as    <a href="ami_1.html#M3" title="AMI_1:mode.3">Instruction-Location</a> of  <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  <b>by </b><i><a class="txt" href="scpqsort.html#E5:16_7"><i><font color="Green" title="E41">A118</font></i></a>, <a class="ref" href="ami_1.html#D4" title="AMI_1:def.4">AMI_1:def 4</a></i>;<br/><a NAME="E3:16_7_1"/>
<font color="Maroon" title="c27">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">a</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:16_7_1"/>
contradiction
 <b>by </b><i><a class="ref" href="scmpds_2.html#T53" title="SCMPDS_2:th.53">SCMPDS_2:53</a></i>;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p2">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="scpqsort.html#E3:16"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_7"><i><font color="Green" title="E39">A116</font></i></a>, <a class="txt" href="scpqsort.html#E3:16_7"><i><font color="Green" title="E40">A117</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p0">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p5">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E6:16_7"><i><font color="Green" title="E42">A119</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p5">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p0">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">k</font></span>)</span> <a href="ami_1.html#K34" title="AMI_1:func.34">+*</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K2" title="SCMPDS_4:func.2">Initialized</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K8" title="SCMPDS_4:func.8">stop</a> <span class="p4">(<span class="default"><a href="scmpds_8.html#K3" title="SCMPDS_8:func.3">while&gt;0</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c4">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmp_gcd.html#K1" title="SCMP_GCD:func.1">intpos</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E17:16"><i><font color="Green" title="E29">A107</font></i></a>, <a class="ref" href="scmpds_2.html#T68" title="SCMPDS_2:th.68">SCMPDS_2:68</a></i>
<br/>.= 
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E13:16"><i><font color="Green" title="E26">A104</font></i></a>, <a class="txt" href="scpqsort.html#E2:16_7"><i><font color="Green" title="E39">A116</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E31:16"/>
( <font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c9">g</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c9">g</font> <a href="graph_2.html#R3" title="GRAPH_2:pred.3">is_non_decreasing_on</a> 1,<font color="Maroon" title="c11">n</font> )
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:16"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="scpqsort.html#E13:16"><i><font color="Green" title="E26">A104</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>


</div>
