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

<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>let </b><font color="Maroon" title="c2">lk</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>let </b><font color="Maroon" title="c3">Key</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of <font color="Maroon" title="c2">lk</font>,6, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">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/>







 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></i>
<br/>.= 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E2:68"/><b>then </b><i><font color="Green" title="E52">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></i>;<br/>
<a NAME="E3:68"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_2"/><i><font color="Green" title="E53">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:68_2"/><b>then </b><i><font color="Green" title="E54">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></i>;<br/>
<a NAME="E3:68_2"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:68_2"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:68_2"/><b>then </b><i><font color="Green" title="E55">A4</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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/>
<a NAME="E5:68_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E6:68_2"/><b>then </b><i><font color="Green" title="E56">A5</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:68_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:68_2"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:68_2"/><b>then </b>
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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:68_2"/><b>then </b><i><font color="Green" title="E57">A6</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1_1"/>
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1_1"/><i><font color="Green" title="E58">A7</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c6">ii</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:68_2_1_1_1"/><i><font color="Green" title="E59">A8</font></i>: 
<font color="Maroon" title="c6">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E4:68_2_1_1_1"/><i><font color="Green" title="E60">A9</font></i>: 
<font color="Maroon" title="c6">ii</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:68_2_1_1_1"><i><font color="Green" title="E59">A8</font></i></a></i>;<br/><a NAME="E5:68_2_1_1_1"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="idea_1.html#E1:68_2"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">ii</font> = <font color="Maroon" title="c6">ii</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="idea_1.html#E3:68_2_1_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E7:68_2_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:68_2"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="idea_1.html#E1:68_2_1_1_1"><i><font color="Green" title="E58">A7</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E8:68_2_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 <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="E9:68_2_1_1_1"/><b>then </b><i><font color="Green" title="E61">A10</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E10:68_2_1_1_1"/><i><font color="Green" title="E62">A11</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="idea_1.html#E2:68_2"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E11:68_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:68_2_1_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="idea_1.html#E9:68_2_1_1_1"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:68_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">ii</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></i>;<br/><a NAME="E13:68_2_1_1_1"/><b>then </b><i><font color="Green" title="E63">A12</font></i>: 
<font color="Maroon" title="c7">ii</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> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</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/><a NAME="E14:68_2_1_1_1"/><i><font color="Green" title="E64">A13</font></i>: 
<font color="Maroon" title="c7">ii</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">ii</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">ii</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E3:68_2_1_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="idea_1.html#E10:68_2_1_1_1"><i><font color="Green" title="E62">A11</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E15:68_2_1_1_1"/><i><font color="Green" title="E65">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E2:68_2"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><i><font color="Green" title="E66">A15</font></i>: <font color="Maroon" title="c4">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c7">ii</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E3:68_2_1_1_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="txt" href="idea_1.html#E14:68_2_1_1_1"><i><font color="Green" title="E64">A13</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"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="idea_1.html#E15:68_2_1_1_1"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">ii</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E4:68_2_1_1_1"><i><font color="Green" title="E60">A9</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ii</font>
<b>by </b><i><a class="txt" href="idea_1.html#E13:68_2_1_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E13:68_2_1_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="idea_1.html#E16:68_2_1_1_1"><i><font color="Green" title="E66">A15</font></i></a>, <a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="idea_1.html#E4:68_2"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_1_2"/><i><font color="Green" title="E58">A16</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="idea_1.html#E6:68_2"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
 <a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2:68_2"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="idea_1.html#E9:68_2"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="idea_1.html#E1:68_2_1_1_2"><i><font color="Green" title="E58">A16</font></i></a>, <a class="ref" href="idea_1.html#D18" target="_self" title="IDEA_1:def.18">Def18</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:68_2"/>
<span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:68"/>
 <a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K16" title="IDEA_1:func.16">IDEA_Q</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">Key</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="idea_1.html#K18" title="IDEA_1:func.18">IDEA_Q_F</a> <font color="Maroon" title="c3">Key</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="idea_1.html#E2:68"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div>
