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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<a NAME="E1:11"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T8" target="_self">Th8</a></i>;<br/>
<i><font color="Green">E10</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="jordan3.html#T19">JORDAN3:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="jordan2b.html#T8" target="_self">Th8</a></i>
;<br/>
<a NAME="E3:11"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E4:11"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_2"/><i><font color="Green">E12</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:11_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:11_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2b.html#E2:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:11_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:11_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>hence </b><a NAME="E4:11_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2b.html#E1:11_2"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:11_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E6:11_2"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2b.html#E1:11_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:11_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2b.html#E1:11_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E8:11_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2b.html#E6:11_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_2_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_2_1_1"><b>case </b></a><a NAME="E1:11_2_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan2b.html#E3:11_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
0
<b>by </b><i><a class="txt" href="jordan2b.html#E8:11_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="jordan2b.html#E2:11_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>

;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_2_1_2"><b>case </b></a><a NAME="E1:11_2_2_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:11_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan2b.html#E1:11"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E3:11_2_2_1_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E4:11_2_2_1_2"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2b.html#E1:11_2_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/><span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default">0 <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="jordan2b.html#E3:11_2_2_1_2"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>
;<br/><b>hence </b><a NAME="E6:11_2_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2b.html#E4:11_2_2_1_2"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:11_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:11"/>
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2b.html#E2:11"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan2b.html#E3:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
