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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green">E31</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green">E33</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:40"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:40"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E6:40"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:40"/><i><font color="Green">E37</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E9:40"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E10:40"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan1g.html#E5:40"><i><font color="Green">E35</font></i></a>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/>
<a NAME="E11:40"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1g.html#E2:40"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1g.html#E3:40"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1g.html#E4:40"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E3:40_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E9:40"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:40_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan1g.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1g.html#E4:40"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1g.html#E8:40"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan1g.html#E1:40_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan4.html#T31">JORDAN4:31</a></i>;<br/><a NAME="E5:40_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E10:40"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan1g.html#E11:40"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan5b.html#T33">JORDAN5B:33</a></i>;<br/><a NAME="E6:40_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1g.html#E8:40"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:40_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:40_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E9:40_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E5:40_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E10:40_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E11:40_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="jordan1g.html#E8:40"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E12:40_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="jordan1g.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>hence </b><a NAME="E13:40_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1g.html#E6:40"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1g.html#E2:40"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1g.html#E3:40"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1g.html#E4:40"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E3:40_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E9:40"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:40_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1g.html#E2:40"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan1g.html#E3:40"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1g.html#E8:40"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan1g.html#E1:40_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan4.html#T32">JORDAN4:32</a></i>;<br/><a NAME="E5:40_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E10:40"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan1g.html#E11:40"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan5b.html#T33">JORDAN5B:33</a></i>;<br/><a NAME="E6:40_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan1g.html#E1:40_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E7:40_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E8:40_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan1g.html#E3:40_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E9:40_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E10:40_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E11:40_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E12:40_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>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan1g.html#E3:40"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E13:40_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>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E5:40_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E14:40_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E11:40_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E15:40_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E16:40_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1g.html#E3:40"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1g.html#E6:40"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_3"><b>suppose </b></a><a NAME="E1:40_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_1_3"/><b>then </b>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan1g.html#E1:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1g.html#E2:40"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan4.html#T27">JORDAN4:27</a></i>;<br/><b>hence </b><a NAME="E3:40_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1g.html#E5:40"><i><font color="Green">E35</font></i></a>, <a class="ref" href="sppol_2.html#T12">SPPOL_2:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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