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<b>let </b><font color="Maroon" title="c47">s</font>, <font color="Maroon" title="c48">s1</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ;<br/><b>let </b><font color="Maroon" title="c49">n0</font>, <font color="Maroon" title="c50">n1</font>, <font color="Maroon" title="c51">n2</font>, <font color="Maroon" title="c52">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c53">f</font>, <font color="Maroon" title="c54">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>assume </b><a NAME="E1:39"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c53">f</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c47">s</font>,<font color="Maroon" title="c49">n0</font> &amp; <font color="Maroon" title="c54">f1</font> <a href="scpisort.html#R1" title="SCPISORT:pred.1">is_FinSequence_on</a> <font color="Maroon" title="c48">s1</font>,<font color="Maroon" title="c49">n0</font> )
 ;<br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green" title="E38">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c53">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c54">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">n</font> )
 ;<br/>

<b>assume </b><a NAME="E3:39"/><i><font color="Green" title="E39">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c50">n1</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c51">n2</font> holds <br/><font color="Maroon" title="c48">s1</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="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="Olive" title="b1">i</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E4:39"/><i><font color="Green" title="E40">A4</font></i>: 
( ( <font color="Maroon" title="c48">s1</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="c50">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c50">n1</font></span>)</span> &amp; <font color="Maroon" title="c48">s1</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="c51">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c51">n2</font></span>)</span> ) or ( <font color="Maroon" title="c50">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c51">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c50">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">n</font> &amp; <font color="Maroon" title="c51">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">n</font> &amp; <font color="Maroon" title="c48">s1</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="c50">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c51">n2</font></span>)</span> &amp; <font color="Maroon" title="c48">s1</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="c51">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c50">n1</font></span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( ( <font color="Maroon" title="c48">s1</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="c50">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c50">n1</font></span>)</span> &amp; <font color="Maroon" title="c48">s1</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="c51">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c51">n2</font></span>)</span> ) or ( <font color="Maroon" title="c50">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c51">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c50">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">n</font> &amp; <font color="Maroon" title="c51">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">n</font> &amp; <font color="Maroon" title="c48">s1</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="c50">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c51">n2</font></span>)</span> &amp; <font color="Maroon" title="c48">s1</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="c51">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c50">n1</font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="scpqsort.html#E4:39"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green" title="E41">A5</font></i>: 
( <font color="Maroon" title="c48">s1</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="c50">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c50">n1</font></span>)</span> &amp; <font color="Maroon" title="c48">s1</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="c51">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c51">n2</font></span>)</span> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c55">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon" title="c55">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c52">n</font>
 ;<br/><a NAME="E2:39_1_1_1"/><b>then </b><i><font color="Green" title="E42">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c55">i</font> &amp; <font color="Maroon" title="c55">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c52">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:39_1_1_1"/><b>then </b><i><font color="Green" title="E43">A7</font></i>: 
( <font color="Maroon" title="c49">n0</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="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> &amp; <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">n</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c56">a</font> = <font color="Maroon" title="c55">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS: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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1"/>
( ( <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c50">n1</font> &amp; <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c51">n2</font> ) or  not <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c50">n1</font> or  not <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c51">n2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_1"/><i><font color="Green" title="E44">A8</font></i>: 
( <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c50">n1</font> &amp; <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c51">n2</font> )
 ;<br/><div class="add"><b>thus </b><font color="Maroon" title="c54">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c55">i</font> = 
<font color="Maroon" title="c48">s1</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="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c56">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c47">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="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c56">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E3:39"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="scpqsort.html#E3:39_1_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_1_1_1_1"><i><font color="Green" title="E44">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c53">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c55">i</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><i><font color="Green" title="E42">A6</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="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2"/><i><font color="Green" title="E44">A9</font></i>: 
(  not <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c50">n1</font> or  not <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c51">n2</font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">n1</font> or <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">n2</font> )
 </span><b>by </b><i><a class="txt" href="scpqsort.html#E1:39_1_1_1_1_2"><i><font color="Green" title="E44">A9</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">n1</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c54">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c55">i</font> = 
<font color="Maroon" title="c47">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="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c56">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c53">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c55">i</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c55">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">n2</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c54">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c55">i</font> = 
<font color="Maroon" title="c47">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="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c56">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c53">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c55">i</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:39_1_1"/>
<font color="Maroon" title="c53">f</font>,<font color="Maroon" title="c54">f1</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green" title="E41">A10</font></i>: 
( <font color="Maroon" title="c50">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c51">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c50">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">n</font> &amp; <font color="Maroon" title="c51">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">n</font> &amp; <font color="Maroon" title="c48">s1</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="c50">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c51">n2</font></span>)</span> &amp; <font color="Maroon" title="c48">s1</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="c51">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">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="c50">n1</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:39_1_2"/><b>then </b><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c50">n1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c49">n0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <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 reconsider </b><font color="Maroon" title="c55">m1</font> = <font color="Maroon" title="c50">n1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c49">n0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS: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="E4:39_1_2"/><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c51">n2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c49">n0</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#E1:39_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c56">m2</font> = <font color="Maroon" title="c51">n2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c49">n0</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS: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="E6:39_1_2"/><i><font color="Green" title="E44">A13</font></i>: 
<font color="Maroon" title="c55">m1</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="c53">f</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E7:39_1_2"/><i><font color="Green" title="E45">A14</font></i>: 
<font color="Maroon" title="c56">m2</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="c54">f1</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E8:39_1_2"/><i><font color="Green" title="E46">A15</font></i>: 
<font color="Maroon" title="c56">m2</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="c53">f</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E9:39_1_2"/><i><font color="Green" title="E47">A16</font></i>: 
<font color="Maroon" title="c50">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n0</font>
 ;<br/><a NAME="E10:39_1_2"/><i><font color="Green" title="E48">A17</font></i>: 
<font color="Maroon" title="c51">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c56">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n0</font>
 ;<br/><i><font color="Green" title="E49">A18</font></i>: <font color="Maroon" title="c53">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c55">m1</font> = 
<font color="Maroon" title="c48">s1</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="c51">n2</font></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_2"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="scpqsort.html#E6:39_1_2"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="scpqsort.html#E9:39_1_2"><i><font color="Green" title="E47">A16</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c54">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c56">m2</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E4:39_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="scpqsort.html#E7:39_1_2"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="scpqsort.html#E10:39_1_2"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
;<br/><i><font color="Green" title="E50">A19</font></i>: <font color="Maroon" title="c53">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c56">m2</font> = 
<font color="Maroon" title="c48">s1</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="c50">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="scpqsort.html#E4:39_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="scpqsort.html#E8:39_1_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="scpqsort.html#E10:39_1_2"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c54">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c55">m1</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_2"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="scpqsort.html#E6:39_1_2"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="scpqsort.html#E9:39_1_2"><i><font color="Green" title="E47">A16</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c57">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:39_1_2_3"/><i><font color="Green" title="E51">A20</font></i>: 
( <font color="Maroon" title="c57">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c55">m1</font> &amp; <font color="Maroon" title="c57">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c56">m2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c57">k</font> &amp; <font color="Maroon" title="c57">k</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="c53">f</font> )
 ;<br/><a NAME="E2:39_1_2_3"/><b>then </b><i><font color="Green" title="E52">A21</font></i>: 
( <font color="Maroon" title="c57">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c55">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n0</font> &amp; <font color="Maroon" title="c57">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c56">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n0</font> )
 ;<br/><a NAME="E3:39_1_2_3"/><i><font color="Green" title="E53">A22</font></i>: 
<font color="Maroon" title="c49">n0</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="c49">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c57">k</font>
 <b>by </b><i><a class="txt" href="scpqsort.html#E1:39_1_2_3"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>thus </b><font color="Maroon" title="c53">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c57">k</font> = 
<font color="Maroon" title="c47">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="c57">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2_3"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="scpisort.html#D1" title="SCPISORT:def.1">SCPISORT:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c48">s1</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="c57">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c49">n0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scpqsort.html#E3:39"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_2_3"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="scpqsort.html#E3:39_1_2_3"><i><font color="Green" title="E53">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c54">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c57">k</font>
<b>by </b><i><a class="txt" href="scpqsort.html#E1:39"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E1:39_1_2_3"><i><font color="Green" title="E51">A20</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="E14:39_1_2"/>
<font color="Maroon" title="c53">f</font>,<font color="Maroon" title="c54">f1</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="scpqsort.html#E2:39"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="scpqsort.html#E2:39_1_2"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="scpqsort.html#E4:39_1_2"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="scpqsort.html#E6:39_1_2"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="scpqsort.html#E8:39_1_2"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="scpqsort.html#E11:39_1_2"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="scpqsort.html#E12:39_1_2"><i><font color="Green" title="E50">A19</font></i></a>, <a class="ref" href="scpisort.html#T4" title="SCPISORT:th.4">SCPISORT:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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