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

<b>reconsider </b><font color="Maroon" title="c3">n2</font> = <font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c4">m2</font> = <font color="Maroon" title="c2">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prgcor_1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c5">ssm</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 1;<br/><a NAME="E2:9_1_1_1_1"/><i><font color="Green" title="E10">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">ssm</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 1 as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E4:9_1_1_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">ssm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="prgcor_1.html#E2:9_1_1_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:9_1_1_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">ssm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E6:9_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">ssm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">ssm</font>
 ;<br/>

<a NAME="E2:9_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E5:9_1_1_1_1"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>
<a NAME="E3:9_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1_1_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">ssm</font> = <font color="Maroon" title="c6">ssm</font> as    <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>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c8">ssn</font> = <font color="Maroon" title="c7">ssm</font>;<br/><b>set </b><font color="Maroon" title="c9">ppn</font> = <font color="Maroon" title="c7">ssm</font>;<br/><a NAME="E8:9_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ssm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ssm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ssm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> implies <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> implies ( <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  ex <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( 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> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> holds <br/>( <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; not <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> ) ) &amp; <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b2">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 1 <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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/>( ( <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">j</font></span>)</span> implies ( <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">j</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">j</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"><font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) ) &amp; ( not <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">j</font></span>)</span> implies ( <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 ) ) ) ) &amp; <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 ) ) ) )
 <b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="prgcor_1.html#E4:9_1_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="pre_ff.html#T4" title="PRE_FF:th.4">PRE_FF:4</a></i>;<br/><b>hence </b><a NAME="E9:9_1_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b2">sm</font>, <font color="Olive" title="b3">sn</font>, <font color="Olive" title="b4">pn</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/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">sm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">sn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">pn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> implies ( <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  ex <font color="Olive" title="b5">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b6">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">k</font> &amp; <font color="Olive" title="b6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">i</font> holds <br/>( <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; not <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> ) ) &amp; <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b6">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">j</font> &amp; <font color="Olive" title="b6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> holds <br/>( ( <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> implies ( <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</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"><font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) ) &amp; ( not <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> implies ( <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 ) ) ) ) &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 ) ) ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_1_2"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="prgcor_1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/><a NAME="E3:9_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">n2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:9_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E10">A7</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <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>  = <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>;<br/><a NAME="E6:9_1_1_1_2"/>
 ex <font color="Olive" title="b1">ssm</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">ssm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; ( for <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">k2</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i><b>then consider </b><font color="Maroon" title="c5">ssm</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E8:9_1_1_1_2"/><i><font color="Green" title="E11">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">ssm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c5">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> ) )
 ;<br/><a NAME="E9:9_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">ssm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1_1_2_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">ssm</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_1_2_1"/><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c7">x</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="c5">ssm</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:9_1_1_1_2_1"/><i><font color="Green" title="E13">A10</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">ssm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">ssm</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_1"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/>
<a NAME="E6:9_1_1_1_2_1"/>
<font color="Maroon" title="c5">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="prgcor_1.html#E4:9_1_1_1_2_1"><i><font color="Green" title="E13">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:9_1_1_1_2_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c6">ssm</font> = <font color="Maroon" title="c5">ssm</font> as    <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>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <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>  = <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>;<br/><a NAME="E11:9_1_1_1_2"/>
 ex <font color="Olive" title="b1">ssn</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">ssn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; ( for <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">k2</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i><b>then consider </b><font color="Maroon" title="c7">ssn</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E13:9_1_1_1_2"/><i><font color="Green" title="E12">A11</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ssn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c7">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> ) )
 ;<br/><a NAME="E14:9_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">ssn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1_1_2_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">ssn</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_1_2_2"/><i><font color="Green" title="E13">A12</font></i>: 
( <font color="Maroon" title="c9">x</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="c7">ssn</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:9_1_1_1_2_2"/><i><font color="Green" title="E14">A13</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">ssn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ssn</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">n3</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_2"><i><font color="Green" title="E13">A12</font></i></a></i>;<br/>
<a NAME="E6:9_1_1_1_2_2"/>
<font color="Maroon" title="c7">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n3</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E13:9_1_1_1_2"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="prgcor_1.html#E4:9_1_1_1_2_2"><i><font color="Green" title="E14">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:9_1_1_1_2_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c8">ssn</font> = <font color="Maroon" title="c7">ssn</font> as    <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>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <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>  = <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>;<br/><a NAME="E16:9_1_1_1_2"/>
 ex <font color="Olive" title="b1">ppn</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">ppn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; ( for <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b2">k2</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i><b>then consider </b><font color="Maroon" title="c9">ppn</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E18:9_1_1_1_2"/><i><font color="Green" title="E13">A14</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">ppn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; ( for <font color="Olive" title="b1">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c9">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> ) )
 ;<br/><a NAME="E19:9_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">ppn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1_1_2_3"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">ppn</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_1_2_3"/><i><font color="Green" title="E14">A15</font></i>: 
( <font color="Maroon" title="c11">x</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">ppn</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:9_1_1_1_2_3"/><i><font color="Green" title="E15">A16</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">ppn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">ppn</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">n3</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_3"><i><font color="Green" title="E14">A15</font></i></a></i>;<br/>
<a NAME="E6:9_1_1_1_2_3"/>
<font color="Maroon" title="c9">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">n3</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E18:9_1_1_1_2"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_3"><i><font color="Green" title="E14">A15</font></i></a>, <a class="txt" href="prgcor_1.html#E4:9_1_1_1_2_3"><i><font color="Green" title="E15">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:9_1_1_1_2_3"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_3"><i><font color="Green" title="E14">A15</font></i></a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">ppn</font> = <font color="Maroon" title="c9">ppn</font> as    <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>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><i><font color="Green" title="E14">A17</font></i>: <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> 0</span>)</span> = 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon" title="c4">m2</font>

;<br/><i><font color="Green" title="E15">A18</font></i>: <font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_1.html#E5:9_1_1_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="prgcor_1.html#E18:9_1_1_1_2"><i><font color="Green" title="E13">A14</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">m2</font>
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a>, <a class="txt" href="prgcor_1.html#E21:9_1_1_1_2"><i><font color="Green" title="E14">A17</font></i></a></i>
;<br/><i><font color="Green" title="E16">A19</font></i>: <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_1.html#E5:9_1_1_1_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font>

;<br/><b>consider </b><font color="Maroon" title="c11">ii0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E25:9_1_1_1_2"/><i><font color="Green" title="E17">A20</font></i>: 
( ( for <font color="Olive" title="b1">k2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">ii0</font> holds <br/><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> ) &amp; <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c11">ii0</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n2</font> )
 <b>by </b><i><a class="txt" href="prgcor_1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="prgcor_1.html#T6" target="_self" title="PRGCOR_1:th.6">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">i0</font> = <font color="Maroon" title="c11">ii0</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_2_7"/>
<font color="Maroon" title="c12">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:9_1_1_1_2_7"/><b>then </b>
<font color="Maroon" title="c4">m2</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="c3">n2</font>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E25:9_1_1_1_2"><i><font color="Green" title="E17">A20</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_1_2_7"/>
contradiction
 <b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E28:9_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">ii0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E29:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E18">A21</font></i>: 
<font color="Maroon" title="c11">ii0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E30:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E19">A22</font></i>: 
<font color="Maroon" title="c12">i0</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> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><div><i><font color="Green" title="E20">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_2_8"/>
<font color="Maroon" title="c12">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n2</font>
 ;<br/><a NAME="E2:9_1_1_1_2_8"/><b>then </b><i><font color="Green" title="E21">A24</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E25:9_1_1_1_2"><i><font color="Green" title="E17">A20</font></i></a></i>;<br/><a NAME="E3:9_1_1_1_2_8"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m2</font>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:9_1_1_1_2_8"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">n2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">n2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E5:9_1_1_1_2_8"/><b>then </b><i><font color="Green" title="E22">A25</font></i>: 
2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E2:9_1_1_1_2_8"><i><font color="Green" title="E21">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="E6:9_1_1_1_2_8"/>
<font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">n2</font>
 <b>by </b><i><a class="ref" href="heine.html#T7" title="HEINE:th.7">HEINE:7</a></i>;<br/><b>hence </b><a NAME="E7:9_1_1_1_2_8"/>
contradiction
 <b>by </b><i><a class="txt" href="prgcor_1.html#E5:9_1_1_1_2_8"><i><font color="Green" title="E22">A25</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><a NAME="E32:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E21">A26</font></i>: 
<font color="Maroon" title="c12">i0</font> <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="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E33:9_1_1_1_2"/>
<font color="Maroon" title="c12">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="prgcor_1.html#E31:9_1_1_1_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E34:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E22">A27</font></i>: 
<font color="Maroon" title="c11">ii0</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E29:9_1_1_1_2"><i><font color="Green" title="E18">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E35:9_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="prgcor_1.html#E29:9_1_1_1_2"><i><font color="Green" title="E18">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E36:9_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E32:9_1_1_1_2"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E37:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E23">A28</font></i>: 
<font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</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
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="prgcor_1.html#E30:9_1_1_1_2"><i><font color="Green" title="E19">A22</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/><a NAME="E39:9_1_1_1_2"/><b>then </b>
2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><b>then </b><i><font color="Green" title="E24">A29</font></i>: <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2
<b>by </b><i><a class="txt" href="prgcor_1.html#E37:9_1_1_1_2"><i><font color="Green" title="E23">A28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="prgcor_1.html#E34:9_1_1_1_2"><i><font color="Green" title="E22">A27</font></i></a></i>
;<br/><a NAME="E41:9_1_1_1_2"/><i><font color="Green" title="E25">A30</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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> holds <br/>( <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; not <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:9_1_1_1_2_11"/><i><font color="Green" title="E26">A31</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> )
 ;<br/>

<a NAME="E2:9_1_1_1_2_11"/><b>then </b><i><font color="Green" title="E27">A32</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E31:9_1_1_1_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:9_1_1_1_2_11"/><i><font color="Green" title="E28">A33</font></i>: 
<font color="Maroon" title="c13">k</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> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_11"><i><font color="Green" title="E26">A31</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E4:9_1_1_1_2_11"/><i><font color="Green" title="E29">A34</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_11"><i><font color="Green" title="E26">A31</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:9_1_1_1_2_11"/><i><font color="Green" title="E30">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E6:9_1_1_1_2_11"/>
<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E2:9_1_1_1_2_11"><i><font color="Green" title="E27">A32</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E7:9_1_1_1_2_11"/><b>then </b>
<font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E4:9_1_1_1_2_11"><i><font color="Green" title="E29">A34</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:9_1_1_1_2_11"/><b>then </b><i><font color="Green" title="E31">A36</font></i>: 
<font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/>
<a NAME="E9:9_1_1_1_2_11"/>
<font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E2:9_1_1_1_2_11"><i><font color="Green" title="E27">A32</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E10:9_1_1_1_2_11"/><b>then </b>
<font color="Maroon" title="c13">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_11"><i><font color="Green" title="E26">A31</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:9_1_1_1_2_11"/><b>then </b><i><font color="Green" title="E32">A37</font></i>: 
<font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">k</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
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_11"><i><font color="Green" title="E28">A33</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/>
<a NAME="E13:9_1_1_1_2_11"/><b>then </b>
2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/>
<b>hence </b><a NAME="E14:9_1_1_1_2_11"/>
( <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; not <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="prgcor_1.html#E25:9_1_1_1_2"><i><font color="Green" title="E17">A20</font></i></a>, <a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_11"><i><font color="Green" title="E26">A31</font></i></a>, <a class="txt" href="prgcor_1.html#E5:9_1_1_1_2_11"><i><font color="Green" title="E30">A35</font></i></a>, <a class="txt" href="prgcor_1.html#E8:9_1_1_1_2_11"><i><font color="Green" title="E31">A36</font></i></a>, <a class="txt" href="prgcor_1.html#E11:9_1_1_1_2_11"><i><font color="Green" title="E32">A37</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E42:9_1_1_1_2"/><i><font color="Green" title="E26">A38</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> holds <br/>( <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; not <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">k</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:9_1_1_1_2_12"/><i><font color="Green" title="E27">A39</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> )
 ;<br/>

<a NAME="E2:9_1_1_1_2_12"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">kk</font> = <font color="Maroon" title="c13">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="E4:9_1_1_1_2_12"/>
( <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">kk</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; not <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E41:9_1_1_1_2"><i><font color="Green" title="E25">A30</font></i></a>, <a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_12"><i><font color="Green" title="E27">A39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1_1_2_12"/>
( <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; not <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E43:9_1_1_1_2"/><i><font color="Green" title="E27">A40</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">ii0</font>
 <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="E44:9_1_1_1_2"/><i><font color="Green" title="E28">A41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ii0</font>
 <b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E45:9_1_1_1_2"/>
<font color="Maroon" title="c12">i0</font> <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="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="prgcor_1.html#E31:9_1_1_1_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E46:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E29">A42</font></i>: 
<font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E43:9_1_1_1_2"><i><font color="Green" title="E27">A40</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E47:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E30">A43</font></i>: 
<font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="prgcor_1.html#E25:9_1_1_1_2"><i><font color="Green" title="E17">A20</font></i></a>, <a class="txt" href="prgcor_1.html#E44:9_1_1_1_2"><i><font color="Green" title="E28">A41</font></i></a></i>;<br/><a NAME="E48:9_1_1_1_2"/>
<font color="Maroon" title="c12">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="prgcor_1.html#E31:9_1_1_1_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E49:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E31">A44</font></i>: 
<font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E50:9_1_1_1_2"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E51:9_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E49:9_1_1_1_2"><i><font color="Green" title="E31">A44</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E52:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E32">A45</font></i>: 
<font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E13:9_1_1_1_2"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">k5</font> = <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E54:9_1_1_1_2"/><i><font color="Green" title="E33">A46</font></i>: 
<font color="Maroon" title="c13">k5</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">n2</font>
 <b>by </b><i><a class="txt" href="prgcor_1.html#E25:9_1_1_1_2"><i><font color="Green" title="E17">A20</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E55:9_1_1_1_2"/>
<font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="prgcor_1.html#E25:9_1_1_1_2"><i><font color="Green" title="E17">A20</font></i></a>, <a class="txt" href="prgcor_1.html#E44:9_1_1_1_2"><i><font color="Green" title="E28">A41</font></i></a>, <a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/><a NAME="E56:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E34">A47</font></i>: 
( <font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="prgcor_1.html#E18:9_1_1_1_2"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="prgcor_1.html#E46:9_1_1_1_2"><i><font color="Green" title="E29">A42</font></i></a>, <a class="txt" href="prgcor_1.html#E52:9_1_1_1_2"><i><font color="Green" title="E32">A45</font></i></a>, <a class="txt" href="prgcor_1.html#E54:9_1_1_1_2"><i><font color="Green" title="E33">A46</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>;<br/><a NAME="E57:9_1_1_1_2"/>
for <font color="Olive" title="b1">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="c12">i0</font> holds <br/>( ( <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">j</font></span>)</span> implies ( <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">j</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"><font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) ) &amp; ( not <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">j</font></span>)</span> implies ( <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:9_1_1_1_2_13"/><i><font color="Green" title="E35">A48</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i0</font> )
 ;<br/>

<a NAME="E2:9_1_1_1_2_13"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">jj</font> = <font color="Maroon" title="c14">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="E4:9_1_1_1_2_13"/><i><font color="Green" title="E36">A49</font></i>: 
<font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E5:9_1_1_1_2_13"/><b>then </b><i><font color="Green" title="E37">A50</font></i>: 
<font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E6:9_1_1_1_2_13"/><i><font color="Green" title="E38">A51</font></i>: 
<font color="Maroon" title="c14">j</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> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<b>thus </b><a NAME="E7:9_1_1_1_2_13"/>
( <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> implies ( <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</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"><font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_1_1_2_13_1"/><i><font color="Green" title="E39">A52</font></i>: 
<font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span>
 ;<br/>

<a NAME="E2:9_1_1_1_2_13_1"/><i><font color="Green" title="E40">A53</font></i>: 
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E6:9_1_1_1_2_13"><i><font color="Green" title="E38">A51</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E3:9_1_1_1_2_13_1"/><i><font color="Green" title="E41">A54</font></i>: 
<font color="Maroon" title="c14">j</font> <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="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:9_1_1_1_2_13_1"/>
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<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="E5:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E42">A55</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_13_1"><i><font color="Green" title="E41">A54</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:9_1_1_1_2_13_1"/>
<font color="Maroon" title="c11">ii0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:9_1_1_1_2_13_1"/><b>then </b>
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E43">A56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E9:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E44">A57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E10:9_1_1_1_2_13_1"/><i><font color="Green" title="E45">A58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2_13_1"><i><font color="Green" title="E43">A56</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E11:9_1_1_1_2_13_1"/><i><font color="Green" title="E46">A59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2_13_1"><i><font color="Green" title="E43">A56</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E12:9_1_1_1_2_13_1"/>
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E13:9_1_1_1_2_13_1"/><b>then </b>
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E5:9_1_1_1_2_13_1"><i><font color="Green" title="E42">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="E14:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E47">A60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E15:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E48">A61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <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="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E2:9_1_1_1_2_13_1"><i><font color="Green" title="E40">A53</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E16:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E49">A62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E17:9_1_1_1_2_13_1"/><i><font color="Green" title="E50">A63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E2:9_1_1_1_2_13_1"><i><font color="Green" title="E40">A53</font></i></a>, <a class="txt" href="prgcor_1.html#E14:9_1_1_1_2_13_1"><i><font color="Green" title="E47">A60</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E18:9_1_1_1_2_13_1"/>
<font color="Maroon" title="c11">ii0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E19:9_1_1_1_2_13_1"/><b>then </b>
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:9_1_1_1_2_13_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E21:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E51">A64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E22:9_1_1_1_2_13_1"/>
<font color="Maroon" title="c12">i0</font> <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="c3">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E31:9_1_1_1_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E23:9_1_1_1_2_13_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</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="E24:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E52">A65</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E21:9_1_1_1_2_13_1"><i><font color="Green" title="E51">A64</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E25:9_1_1_1_2_13_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="txt" href="prgcor_1.html#E15:9_1_1_1_2_13_1"><i><font color="Green" title="E48">A61</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E26:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E53">A66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="txt" href="prgcor_1.html#E15:9_1_1_1_2_13_1"><i><font color="Green" title="E48">A61</font></i></a>, <a class="txt" href="prgcor_1.html#E24:9_1_1_1_2_13_1"><i><font color="Green" title="E52">A65</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E27:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E54">A67</font></i>: 
<font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E13:9_1_1_1_2"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<a NAME="E28:9_1_1_1_2_13_1"/><i><font color="Green" title="E55">A68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2_13_1"><i><font color="Green" title="E43">A56</font></i></a>, <a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E29:9_1_1_1_2_13_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <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> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E30:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E56">A69</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E31:9_1_1_1_2_13_1"/><i><font color="Green" title="E57">A70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E32:9_1_1_1_2_13_1"/>
<font color="Maroon" title="c12">i0</font> <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="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E31:9_1_1_1_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E33:9_1_1_1_2_13_1"/><b>then </b>
<font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E31:9_1_1_1_2_13_1"><i><font color="Green" title="E57">A70</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E58">A71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E28:9_1_1_1_2_13_1"><i><font color="Green" title="E55">A68</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E35:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E59">A72</font></i>: 
<font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E13:9_1_1_1_2"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<a NAME="E36:9_1_1_1_2_13_1"/><i><font color="Green" title="E60">A73</font></i>: 
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E6:9_1_1_1_2_13"><i><font color="Green" title="E38">A51</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E37:9_1_1_1_2_13_1"/><i><font color="Green" title="E61">A74</font></i>: 
<font color="Maroon" title="c14">j</font> <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="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E38:9_1_1_1_2_13_1"/>
<font color="Maroon" title="c15">jj</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E39:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E62">A75</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E37:9_1_1_1_2_13_1"><i><font color="Green" title="E61">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="E40:9_1_1_1_2_13_1"/>
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">jj</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E41:9_1_1_1_2_13_1"/><b>then </b>
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E39:9_1_1_1_2_13_1"><i><font color="Green" title="E62">A75</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E42:9_1_1_1_2_13_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E43:9_1_1_1_2_13_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <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="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="txt" href="prgcor_1.html#E36:9_1_1_1_2_13_1"><i><font color="Green" title="E60">A73</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E44:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E63">A76</font></i>: 
<font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E27:9_1_1_1_2_13_1"><i><font color="Green" title="E54">A67</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E45:9_1_1_1_2_13_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <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> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="txt" href="prgcor_1.html#E10:9_1_1_1_2_13_1"><i><font color="Green" title="E45">A58</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<b>then </b><i><font color="Green" title="E64">A77</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font>
<b>by </b><i><a class="txt" href="prgcor_1.html#E4:9_1_1_1_2_13"><i><font color="Green" title="E36">A49</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/>
<i><font color="Green" title="E65">A78</font></i>: <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_1.html#E5:9_1_1_1_2_13"><i><font color="Green" title="E37">A50</font></i></a>, <a class="txt" href="prgcor_1.html#E8:9_1_1_1_2_13_1"><i><font color="Green" title="E43">A56</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></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="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E48:9_1_1_1_2_13_1"/><i><font color="Green" title="E66">A79</font></i>: 
<font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="prgcor_1.html#E34:9_1_1_1_2_13_1"><i><font color="Green" title="E58">A71</font></i></a>, <a class="txt" href="prgcor_1.html#E46:9_1_1_1_2_13_1"><i><font color="Green" title="E64">A77</font></i></a></i>;<br/>
<a NAME="E49:9_1_1_1_2_13_1"/>
2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="card_4.html#T51" title="CARD_4:th.51">CARD_4:51</a></i>;<br/>
<a NAME="E50:9_1_1_1_2_13_1"/><b>then </b>
2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E51:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E67">A80</font></i>: 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E52:9_1_1_1_2_13_1"/><b>then </b><i><font color="Green" title="E68">A81</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13_1"><i><font color="Green" title="E39">A52</font></i></a>, <a class="txt" href="prgcor_1.html#E11:9_1_1_1_2_13_1"><i><font color="Green" title="E46">A59</font></i></a>, <a class="txt" href="prgcor_1.html#E15:9_1_1_1_2_13_1"><i><font color="Green" title="E48">A61</font></i></a>, <a class="txt" href="prgcor_1.html#E44:9_1_1_1_2_13_1"><i><font color="Green" title="E63">A76</font></i></a>, <a class="txt" href="prgcor_1.html#E47:9_1_1_1_2_13_1"><i><font color="Green" title="E65">A78</font></i></a>, <a class="txt" href="prgcor_1.html#E48:9_1_1_1_2_13_1"><i><font color="Green" title="E66">A79</font></i></a>, <a class="ref" href="prgcor_1.html#T2" target="_self" title="PRGCOR_1:th.2">Th2</a></i>;<br/>
<font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> = 
<font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_1.html#E18:9_1_1_1_2"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="prgcor_1.html#E34:9_1_1_1_2_13_1"><i><font color="Green" title="E58">A71</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</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#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13_1"><i><font color="Green" title="E39">A52</font></i></a>, <a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="txt" href="prgcor_1.html#E16:9_1_1_1_2_13_1"><i><font color="Green" title="E49">A62</font></i></a>, <a class="txt" href="prgcor_1.html#E17:9_1_1_1_2_13_1"><i><font color="Green" title="E50">A63</font></i></a>, <a class="txt" href="prgcor_1.html#E27:9_1_1_1_2_13_1"><i><font color="Green" title="E54">A67</font></i></a>, <a class="txt" href="prgcor_1.html#E46:9_1_1_1_2_13_1"><i><font color="Green" title="E64">A77</font></i></a>, <a class="txt" href="prgcor_1.html#E47:9_1_1_1_2_13_1"><i><font color="Green" title="E65">A78</font></i></a>, <a class="txt" href="prgcor_1.html#E48:9_1_1_1_2_13_1"><i><font color="Green" title="E66">A79</font></i></a>, <a class="txt" href="prgcor_1.html#E51:9_1_1_1_2_13_1"><i><font color="Green" title="E67">A80</font></i></a>, <a class="ref" href="prgcor_1.html#T3" target="_self" title="PRGCOR_1:th.3">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="prgcor_1.html#E18:9_1_1_1_2"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="prgcor_1.html#E26:9_1_1_1_2_13_1"><i><font color="Green" title="E53">A66</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E54:9_1_1_1_2_13_1"/>
( <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</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"><font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 <b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_1"><i><font color="Green" title="E44">A57</font></i></a>, <a class="txt" href="prgcor_1.html#E15:9_1_1_1_2_13_1"><i><font color="Green" title="E48">A61</font></i></a>, <a class="txt" href="prgcor_1.html#E30:9_1_1_1_2_13_1"><i><font color="Green" title="E56">A69</font></i></a>, <a class="txt" href="prgcor_1.html#E35:9_1_1_1_2_13_1"><i><font color="Green" title="E59">A72</font></i></a>, <a class="txt" href="prgcor_1.html#E52:9_1_1_1_2_13_1"><i><font color="Green" title="E68">A81</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:9_1_1_1_2_13"/>
( not <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> implies ( <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2_13_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_1_1_2_13_2"/><i><font color="Green" title="E39">A82</font></i>: 
not <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span>
 ;<br/>

<a NAME="E2:9_1_1_1_2_13_2"/><i><font color="Green" title="E40">A83</font></i>: 
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E6:9_1_1_1_2_13"><i><font color="Green" title="E38">A51</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E3:9_1_1_1_2_13_2"/><i><font color="Green" title="E41">A84</font></i>: 
<font color="Maroon" title="c14">j</font> <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="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:9_1_1_1_2_13_2"/>
<font color="Maroon" title="c15">jj</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E42">A85</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E3:9_1_1_1_2_13_2"><i><font color="Green" title="E41">A84</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:9_1_1_1_2_13_2"/>
<font color="Maroon" title="c11">ii0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:9_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E43">A86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E9:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E44">A87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E10:9_1_1_1_2_13_2"/><i><font color="Green" title="E45">A88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2_13_2"><i><font color="Green" title="E43">A86</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E11:9_1_1_1_2_13_2"/>
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">jj</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E12:9_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E5:9_1_1_1_2_13_2"><i><font color="Green" title="E42">A85</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E46">A89</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E14:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E47">A90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <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="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E2:9_1_1_1_2_13_2"><i><font color="Green" title="E40">A83</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E15:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E48">A91</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_2"><i><font color="Green" title="E44">A87</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E16:9_1_1_1_2_13_2"/>
<font color="Maroon" title="c11">ii0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E17:9_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:9_1_1_1_2_13_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E19:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E49">A92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E20:9_1_1_1_2_13_2"/>
<font color="Maroon" title="c12">i0</font> <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="c3">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E31:9_1_1_1_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E21:9_1_1_1_2_13_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</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="E22:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E50">A93</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E19:9_1_1_1_2_13_2"><i><font color="Green" title="E49">A92</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E23:9_1_1_1_2_13_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_2"><i><font color="Green" title="E44">A87</font></i></a>, <a class="txt" href="prgcor_1.html#E14:9_1_1_1_2_13_2"><i><font color="Green" title="E47">A90</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E24:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E51">A94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_2"><i><font color="Green" title="E44">A87</font></i></a>, <a class="txt" href="prgcor_1.html#E14:9_1_1_1_2_13_2"><i><font color="Green" title="E47">A90</font></i></a>, <a class="txt" href="prgcor_1.html#E22:9_1_1_1_2_13_2"><i><font color="Green" title="E50">A93</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E25:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E52">A95</font></i>: 
<font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E13:9_1_1_1_2"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<a NAME="E26:9_1_1_1_2_13_2"/><i><font color="Green" title="E53">A96</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2_13_2"><i><font color="Green" title="E43">A86</font></i></a>, <a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_2"><i><font color="Green" title="E44">A87</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E27:9_1_1_1_2_13_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <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> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E28:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E54">A97</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_2"><i><font color="Green" title="E44">A87</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E29:9_1_1_1_2_13_2"/><i><font color="Green" title="E55">A98</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E30:9_1_1_1_2_13_2"/>
<font color="Maroon" title="c12">i0</font> <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="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E31:9_1_1_1_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E31:9_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E29:9_1_1_1_2_13_2"><i><font color="Green" title="E55">A98</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E56">A99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E26:9_1_1_1_2_13_2"><i><font color="Green" title="E53">A96</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E33:9_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E13:9_1_1_1_2"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<a NAME="E34:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E57">A100</font></i>: 
<font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E28:9_1_1_1_2_13_2"><i><font color="Green" title="E54">A97</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E35:9_1_1_1_2_13_2"/><i><font color="Green" title="E58">A101</font></i>: 
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E6:9_1_1_1_2_13"><i><font color="Green" title="E38">A51</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E36:9_1_1_1_2_13_2"/><i><font color="Green" title="E59">A102</font></i>: 
<font color="Maroon" title="c14">j</font> <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="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E37:9_1_1_1_2_13_2"/>
<font color="Maroon" title="c15">jj</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">jj</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E38:9_1_1_1_2_13_2"/><b>then </b><i><font color="Green" title="E60">A103</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E36:9_1_1_1_2_13_2"><i><font color="Green" title="E59">A102</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E39:9_1_1_1_2_13_2"/>
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E40:9_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E38:9_1_1_1_2_13_2"><i><font color="Green" title="E60">A103</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E41:9_1_1_1_2_13_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<b>then </b><i><font color="Green" title="E61">A104</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="prgcor_1.html#E35:9_1_1_1_2_13_2"><i><font color="Green" title="E58">A101</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2_13_2"><i><font color="Green" title="E43">A86</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/>
<a NAME="E43:9_1_1_1_2_13_2"/><i><font color="Green" title="E62">A105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E2:9_1_1_1_2_13_2"><i><font color="Green" title="E40">A83</font></i></a>, <a class="txt" href="prgcor_1.html#E13:9_1_1_1_2_13_2"><i><font color="Green" title="E46">A89</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E44:9_1_1_1_2_13_2"/><i><font color="Green" title="E63">A106</font></i>: 
<font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E25:9_1_1_1_2_13_2"><i><font color="Green" title="E52">A95</font></i></a>, <a class="txt" href="prgcor_1.html#E42:9_1_1_1_2_13_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E45:9_1_1_1_2_13_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <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> 0
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13"><i><font color="Green" title="E35">A48</font></i></a>, <a class="txt" href="prgcor_1.html#E10:9_1_1_1_2_13_2"><i><font color="Green" title="E45">A88</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<b>then </b><i><font color="Green" title="E64">A107</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font>
<b>by </b><i><a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_2"><i><font color="Green" title="E44">A87</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font>
<b>by </b><i><a class="txt" href="prgcor_1.html#E4:9_1_1_1_2_13"><i><font color="Green" title="E36">A49</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/>
<i><font color="Green" title="E65">A108</font></i>: <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_1.html#E5:9_1_1_1_2_13"><i><font color="Green" title="E37">A50</font></i></a>, <a class="txt" href="prgcor_1.html#E8:9_1_1_1_2_13_2"><i><font color="Green" title="E43">A86</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></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="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E48:9_1_1_1_2_13_2"/><i><font color="Green" title="E66">A109</font></i>: 
<font color="Maroon" title="c6">ssm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="prgcor_1.html#E32:9_1_1_1_2_13_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="txt" href="prgcor_1.html#E46:9_1_1_1_2_13_2"><i><font color="Green" title="E64">A107</font></i></a></i>;<br/>
<font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> = 
<font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">jj</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_1.html#E18:9_1_1_1_2"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="prgcor_1.html#E32:9_1_1_1_2_13_2"><i><font color="Green" title="E56">A99</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
<b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13_2"><i><font color="Green" title="E39">A82</font></i></a>, <a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_2"><i><font color="Green" title="E44">A87</font></i></a>, <a class="txt" href="prgcor_1.html#E15:9_1_1_1_2_13_2"><i><font color="Green" title="E48">A91</font></i></a>, <a class="txt" href="prgcor_1.html#E25:9_1_1_1_2_13_2"><i><font color="Green" title="E52">A95</font></i></a>, <a class="txt" href="prgcor_1.html#E43:9_1_1_1_2_13_2"><i><font color="Green" title="E62">A105</font></i></a>, <a class="txt" href="prgcor_1.html#E46:9_1_1_1_2_13_2"><i><font color="Green" title="E64">A107</font></i></a>, <a class="txt" href="prgcor_1.html#E47:9_1_1_1_2_13_2"><i><font color="Green" title="E65">A108</font></i></a>, <a class="txt" href="prgcor_1.html#E48:9_1_1_1_2_13_2"><i><font color="Green" title="E66">A109</font></i></a>, <a class="ref" href="prgcor_1.html#T5" target="_self" title="PRGCOR_1:th.5">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">ii0</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">jj</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2
<b>by </b><i><a class="txt" href="prgcor_1.html#E18:9_1_1_1_2"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="prgcor_1.html#E24:9_1_1_1_2_13_2"><i><font color="Green" title="E51">A94</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E50:9_1_1_1_2_13_2"/>
( <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">ssn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">ppn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">i0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 )
 <b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2_13_2"><i><font color="Green" title="E39">A82</font></i></a>, <a class="txt" href="prgcor_1.html#E9:9_1_1_1_2_13_2"><i><font color="Green" title="E44">A87</font></i></a>, <a class="txt" href="prgcor_1.html#E14:9_1_1_1_2_13_2"><i><font color="Green" title="E47">A90</font></i></a>, <a class="txt" href="prgcor_1.html#E34:9_1_1_1_2_13_2"><i><font color="Green" title="E57">A100</font></i></a>, <a class="txt" href="prgcor_1.html#E44:9_1_1_1_2_13_2"><i><font color="Green" title="E63">A106</font></i></a>, <a class="txt" href="prgcor_1.html#E47:9_1_1_1_2_13_2"><i><font color="Green" title="E65">A108</font></i></a>, <a class="txt" href="prgcor_1.html#E48:9_1_1_1_2_13_2"><i><font color="Green" title="E66">A109</font></i></a>, <a class="ref" href="prgcor_1.html#T4" target="_self" title="PRGCOR_1:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E58:9_1_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ex <font color="Olive" title="b2">sm</font>, <font color="Olive" title="b3">sn</font>, <font color="Olive" title="b4">pn</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/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">sm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">sn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">pn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( not <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> implies ( <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  ex <font color="Olive" title="b5">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b6">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">k</font> &amp; <font color="Olive" title="b6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">i</font> holds <br/>( <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; not <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> ) ) &amp; <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 &amp; <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b6">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">j</font> &amp; <font color="Olive" title="b6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">i</font> holds <br/>( ( <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> implies ( <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</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"><font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) ) &amp; ( not <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">sm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> implies ( <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">sn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 2 ) ) ) ) &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">pn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 ) ) ) )
 <b>by </b><i><a class="txt" href="prgcor_1.html#E1:9_1_1_1_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="prgcor_1.html#E8:9_1_1_1_2"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="prgcor_1.html#E13:9_1_1_1_2"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="prgcor_1.html#E18:9_1_1_1_2"><i><font color="Green" title="E13">A14</font></i></a>, <a class="txt" href="prgcor_1.html#E22:9_1_1_1_2"><i><font color="Green" title="E15">A18</font></i></a>, <a class="txt" href="prgcor_1.html#E23:9_1_1_1_2"><i><font color="Green" title="E16">A19</font></i></a>, <a class="txt" href="prgcor_1.html#E29:9_1_1_1_2"><i><font color="Green" title="E18">A21</font></i></a>, <a class="txt" href="prgcor_1.html#E31:9_1_1_1_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="txt" href="prgcor_1.html#E40:9_1_1_1_2"><i><font color="Green" title="E24">A29</font></i></a>, <a class="txt" href="prgcor_1.html#E42:9_1_1_1_2"><i><font color="Green" title="E26">A38</font></i></a>, <a class="txt" href="prgcor_1.html#E47:9_1_1_1_2"><i><font color="Green" title="E30">A43</font></i></a>, <a class="txt" href="prgcor_1.html#E56:9_1_1_1_2"><i><font color="Green" title="E34">A47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
