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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="topreal4.html#T3">TOPREAL4:3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green">E31</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:32"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E8:32"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:32_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:32"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</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> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E11:32"/><i><font color="Green">E35</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:32"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:32"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E11:32"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:32"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:32"/><i><font color="Green">E38</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:32"/><i><font color="Green">E39</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E17:32"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E18:32"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E15:32"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E19:32"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E17:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>

;<br/>
<a NAME="E21:32"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E19:32"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E22:32"/><b>then </b><i><font color="Green">E43</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<a NAME="E23:32"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="jordan20.html#E22:32"><i><font color="Green">E43</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E25:32"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<a NAME="E26:32"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E21:32"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E27:32"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E17:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E28:32"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E26:32"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E29:32"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E30:32"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E31:32"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E16:32"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:32"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E16:32"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan20.html#E30:32"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E33:32"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E11:32"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E34:32"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E35:32"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E36:32"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E38:32"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E38:32"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1"><b>case </b></a><a NAME="E1:32_3_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:32_3_1_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E36</font></i></a>, <a class="ref" href="sprect_4.html#T5">SPRECT_4:5</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_1_1"/><i><font color="Green">E58</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_1_1"/>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<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>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_1_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E4:32_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E5:32_3_1_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_1_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:32_3_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R3" target="_self">is_Lout</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>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R4" target="_self">is_Rout</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>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E5:32_3_1_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan20.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:32_3_1_1_1_1_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E3:32_3_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_1_1_1_2"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:32_3_1_1_1_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:32_3_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E36</font></i></a>, <a class="ref" href="sprect_4.html#T5">SPRECT_4:5</a></i>;<br/><a NAME="E8:32_3_1_1_1_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_2_2"/><i><font color="Green">E59</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_2_2"/>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<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>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_2"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E4:32_3_1_1_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_1_2_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E4:32_3_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_2_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_2_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_2_2_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E7:32_3_1_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_1_1_1_2_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_2_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_2_2_1_1_2"/><b>then </b>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan20.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:32_3_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_1_1_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>6</sub></font>,1,<font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E10:32_3_1_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E11:32_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E11:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:32_3_1_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:32_3_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:32_3_1_1_1_1_2_3"/><i><font color="Green">E63</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><a NAME="E4:32_3_1_1_1_1_2_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:32_3_1_1_1_1_2_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:32_3_1_1_1_1_2_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</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> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E6:32_3_1_1_1_1_2_3"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E9:32_3_1_1_1_1_2_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_2_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:32_3_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_2_3"/><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_2_3"/><i><font color="Green">E71</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E15:32_3_1_1_1_1_2_3"/><i><font color="Green">E72</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_1_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T26">JORDAN5C:26</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_2_3"/><i><font color="Green">E73</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_1_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T25">JORDAN5C:25</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_2_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E18:32_3_1_1_1_1_2_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:32_3_1_1_1_1_2_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:32_3_1_1_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan20.html#E13:32_3_1_1_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan20.html#E17:32_3_1_1_1_1_2_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E20:32_3_1_1_1_1_2_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:32_3_1_1_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_2_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_1_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_1_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E78</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E15:32_3_1_1_1_1_2_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_2_3_1_1_1_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_1_1_1_2_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div><a NAME="E4:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E80</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1

;<br/><a NAME="E6:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font>]
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E4:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E8:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E31:32"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E5:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E84</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E16:32_3_1_1_1_1_2_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E9:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_2_3_1_1_1"/>
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Olive">b<sub>1</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E9:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan20.html#E11:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E84</font></i></a></i>;<br/><a NAME="E15:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_2_3_1_1_1_3"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_1_1_1_2_3"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_1_2_3_1_1_1_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_3_1_1_1_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E18:32_3_1_1_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:32_3_1_1_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan20.html#E13:32_3_1_1_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan20.html#E17:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E20:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:32_3_1_1_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E21:32_3_1_1_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E13:32_3_1_1_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E19:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan20.html#E20:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E23:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan20.html#E15:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E24:32_3_1_1_1_1_2_3_1_1_1"/><i><font color="Green">E93</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E16:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/><a NAME="E25:32_3_1_1_1_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E23:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E26:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E27:32_3_1_1_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_1_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_1_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_1_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E23:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan20.html#E24:32_3_1_1_1_1_2_3_1_1_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="jordan20.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E28:32_3_1_1_1_1_2_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2"/><i><font color="Green">E77</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_1"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_1_1_1_2_3"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_1_2_3_1_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_3_1_1_2_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_3_1_1_1_1_2_3_1_1_2"/><i><font color="Green">E79</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Olive">b<sub>1</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_2_3_1_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T23">JORDAN5C:23</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E13:32_3_1_1_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E19:32_3_1_1_1_1_2_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan20.html#E20:32_3_1_1_1_1_2_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan20.html#E4:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E8:32_3_1_1_1_1_2_3_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_1_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T26">JORDAN5C:26</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green">E83</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_2_3_1_1_2_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan5c.html#T23">JORDAN5C:23</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan20.html#E7:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E4:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E84</font></i></a></i>;<br/><a NAME="E8:32_3_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E7:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E86</font></i>: 
not  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E11:32_3_1_1_1_1_2_3_1_1_2_2"/><i><font color="Green">E87</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_2_3_1_1_2_2"/><i><font color="Green">E88</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_2_3_1_1_2_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="topreal1.html#V3">s.n.c.</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E11:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E15:32_3_1_1_1_1_2_3_1_1_2_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E9:32_3_1_1_1_1_2_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E11:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan20.html#E13:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></font> <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="E18:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan20.html#E15:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jordan2c.html#T2">JORDAN2C:2</a></i>;<br/><div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E15:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E3:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_1_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E9:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E13:32_3_1_1_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:32_3_1_1_1_1_2_3_1_1_2_2_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan20.html#E13:32_3_1_1_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan20.html#E7:32_3_1_1_1_1_2_3_1_1_2_2_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E10:32_3_1_1_1_1_2_3_1_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32_3_1_1_1_1_2_3_1_1_2_2_1"><i><font color="Green">E94</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E15:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan20.html#E18:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan20.html#E19:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E93</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_2_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32_3_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_3_1_1_2_2_2_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_3_1_1_2_2_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_2_3_1_1_2_2_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32_3_1_1_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E21:32_3_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E22:32_3_1_1_1_1_2_3_1_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/><b>hence </b><a NAME="E23:32_3_1_1_1_1_2_3_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_1_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_1_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E20:32_3_1_1_1_1_2_3_1_1_2_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="jordan20.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><a NAME="E11:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E10:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font>

;<br/><a NAME="E13:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_1_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E15:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E32:32"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_1_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_2_3_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_1_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E16:32_3_1_1_1_1_2_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><b>hence </b><a NAME="E18:32_3_1_1_1_1_2_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E16:32_3_1_1_1_1_2_3_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E22:32_3_1_1_1_1_2_3"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E15:32_3_1_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_1_2_4"/><i><font color="Green">E63</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_1_2_4"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_2_4_1_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<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><b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E13:32_3_1_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_1_1_1_2_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan20.html#T19" target="_self">Th19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_4_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_4_1_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_2_4_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32_3_1_1_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_4"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_2_4_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_1_1_1_2_4_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_1_2_4_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1_1_1_2_4"><i><font color="Green">E63</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="E4:32_3_1_1_1_1_2_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E16:32_3_1_1_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R3" target="_self">is_Lout</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>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R4" target="_self">is_Rout</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>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan20.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:32_3_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R3" target="_self">is_Lout</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>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R4" target="_self">is_Rout</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>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2"><b>case </b></a><a NAME="E1:32_3_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:32_3_1_2_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:32_3_1_2_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:32_3_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E5:32_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E36</font></i></a>, <a class="ref" href="sprect_4.html#T5">SPRECT_4:5</a></i>;<br/><a NAME="E6:32_3_1_2_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_1_1"/><i><font color="Green">E58</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_1_1"/>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<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>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:32_3_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_1_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E4:32_3_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:32_3_1_2_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:32_3_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_2_1_1_1_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E3:32_3_1_2_1_1_1_1_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E5:32_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_2_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:32_3_1_2_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R3" target="_self">is_Lout</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>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R4" target="_self">is_Rout</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>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E5:32_3_1_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:32_3_1_2_1_1_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E3:32_3_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_2_1_1_2"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:32_3_1_2_1_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:32_3_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E36</font></i></a>, <a class="ref" href="sprect_4.html#T5">SPRECT_4:5</a></i>;<br/><a NAME="E8:32_3_1_2_1_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_2_2"/><i><font color="Green">E59</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_2_2"/>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<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>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:32_3_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_2"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E4:32_3_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:32_3_1_2_1_1_2_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E7:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E4:32_3_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:32_3_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E8:32"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_2_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_2_1_1_2_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E3:32_3_1_2_1_1_2_2_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E7:32_3_1_2_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_2_1_1_2_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_2_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_2_1_1_2_2_1_1_2"/><b>then </b>
 <a href="jordan6.html#K5">Segment</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>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan20.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:32_3_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>6</sub></font>,1,<font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E10:32_3_1_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E11:32_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E11:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan20.html#E28:32"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E12:32_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:32_3_1_2_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E13:32_3_1_2_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:32_3_1_2_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:32_3_1_2_1_1_2_3"/><i><font color="Green">E63</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><a NAME="E4:32_3_1_2_1_1_2_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:32_3_1_2_1_1_2_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:32_3_1_2_1_1_2_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</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> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E6:32_3_1_2_1_1_2_3"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E9:32_3_1_2_1_1_2_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E10:32_3_1_2_1_1_2_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:32_3_1_2_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:32_3_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:32_3_1_2_1_1_2_3"/><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:32_3_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:32_3_1_2_1_1_2_3"/><i><font color="Green">E71</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E16:32_3_1_2_1_1_2_3"/><i><font color="Green">E72</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_2_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T26">JORDAN5C:26</a></i>;<br/><a NAME="E17:32_3_1_2_1_1_2_3"/><i><font color="Green">E73</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>15</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_2_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T25">JORDAN5C:25</a></i>;<br/><a NAME="E18:32_3_1_2_1_1_2_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E19:32_3_1_2_1_1_2_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E20:32_3_1_2_1_1_2_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32_3_1_2_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan20.html#E18:32_3_1_2_1_1_2_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E21:32_3_1_2_1_1_2_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32_3_1_2_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_2_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_2_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E15:32_3_1_2_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E78</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E16:32_3_1_2_1_1_2_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:32_3_1_2_1_1_2_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:32_3_1_2_1_1_2_3_1_1_1_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E13:32_3_1_2_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_2_1_1_2_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div><a NAME="E4:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E80</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1

;<br/><a NAME="E6:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font>]
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E4:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E8:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E31:32"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E5:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E10:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E35:32"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E84</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E17:32_3_1_2_1_1_2_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E12:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E9:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E13:32_3_1_2_1_1_2_3_1_1_1"/>
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Olive">b<sub>1</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E14:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E9:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan20.html#E11:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E84</font></i></a></i>;<br/><a NAME="E15:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E87</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_2_3_1_1_1_3"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E15:32_3_1_2_1_1_2_3"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_2_1_1_2_3_1_1_1_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_3_1_1_1_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E17:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E18:32_3_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32_3_1_2_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan20.html#E17:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E20:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:32_3_1_2_1_1_2_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E21:32_3_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E19:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan20.html#E20:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E23:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan20.html#E15:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E24:32_3_1_2_1_1_2_3_1_1_1"/><i><font color="Green">E93</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E16:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/><a NAME="E25:32_3_1_2_1_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E23:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E26:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E27:32_3_1_2_1_1_2_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_2_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_2_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan20.html#E15:32_3_1_2_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E23:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan20.html#E24:32_3_1_2_1_1_2_3_1_1_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="jordan20.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E28:32_3_1_2_1_1_2_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2"/><i><font color="Green">E77</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_1"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E15:32_3_1_2_1_1_2_3"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_2_1_1_2_3_1_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_3_1_1_2_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:32_3_1_2_1_1_2_3_1_1_2"/><i><font color="Green">E79</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Olive">b<sub>1</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><a NAME="E4:32_3_1_2_1_1_2_3_1_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T23">JORDAN5C:23</a></i>;<br/><a NAME="E5:32_3_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E20:32_3_1_2_1_1_2_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan20.html#E21:32_3_1_2_1_1_2_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan20.html#E4:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E8:32_3_1_2_1_1_2_3_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_2_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T26">JORDAN5C:26</a></i>;<br/><a NAME="E9:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green">E83</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E12:32"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E5:32_3_1_2_1_1_2_3_1_1_2_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan5c.html#T23">JORDAN5C:23</a></i>;<br/><a NAME="E6:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan20.html#E7:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan20.html#E4:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E84</font></i></a></i>;<br/><a NAME="E8:32_3_1_2_1_1_2_3_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/><a NAME="E9:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E7:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E86</font></i>: 
not  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E11:32_3_1_2_1_1_2_3_1_1_2_2"/><i><font color="Green">E87</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E12:32_3_1_2_1_1_2_3_1_1_2_2"/><i><font color="Green">E88</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E13:32_3_1_2_1_1_2_3_1_1_2_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="topreal1.html#V3">s.n.c.</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E14:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E11:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E15:32_3_1_2_1_1_2_3_1_1_2_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E9:32_3_1_2_1_1_2_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E16:32_3_1_2_1_1_2_3_1_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E10:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan20.html#E11:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan20.html#E13:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/><a NAME="E17:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></font> <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="E18:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan20.html#E15:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jordan2c.html#T2">JORDAN2C:2</a></i>;<br/><div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E15:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E3:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E10:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan20.html#E13:32_3_1_2_1_1_2_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E5:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E9:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/><a NAME="E6:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:32_3_1_2_1_1_2_3_1_1_2_2_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E9:32_3_1_2_1_1_2_3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan20.html#E7:32_3_1_2_1_1_2_3_1_1_2_2_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E10:32_3_1_2_1_1_2_3_1_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32_3_1_2_1_1_2_3_1_1_2_2_1"><i><font color="Green">E94</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E15:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan20.html#E18:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="jordan20.html#E19:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E93</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_2_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_2_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32_3_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_3_1_1_2_2_2_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_3_1_1_2_2_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_2_3_1_1_2_2_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E15:32_3_1_2_1_1_2_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E21:32_3_1_2_1_1_2_3_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E22:32_3_1_2_1_1_2_3_1_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/><b>hence </b><a NAME="E23:32_3_1_2_1_1_2_3_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E6:32_3_1_2_1_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_2_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E20:32_3_1_2_1_1_2_3_1_1_2_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="jordan20.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><a NAME="E11:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E10:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font>

;<br/><a NAME="E13:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_2_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E14:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E25:32"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E15:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E16:32_3_1_2_1_1_2_3_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E32:32"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan20.html#E10:32_3_1_2_1_1_2_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jordan5c.html#T24">JORDAN5C:24</a></i>;<br/><a NAME="E17:32_3_1_2_1_1_2_3_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>, <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E3:32_3_1_2_1_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan20.html#E17:32_3_1_2_1_1_2_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><b>hence </b><a NAME="E18:32_3_1_2_1_1_2_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E4:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan20.html#E36:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan20.html#E16:32_3_1_2_1_1_2_3_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E23:32_3_1_2_1_1_2_3"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E15:32_3_1_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st  <a href="jordan5c.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_3_1_2_1_1_2_4"/><i><font color="Green">E63</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:32_3_1_2_1_1_2_4"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_2_1_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_2_1_1_2_4_1_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<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><b>by </b><i><a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E13:32_3_1_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan20.html#E2:32_3_1_2_1_1_2_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan20.html#T19" target="_self">Th19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_4_1_1_1"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_4_1_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_2_4_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:32_3_1_2_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_4"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_2_1_1_2_4_1_1_2"><b>case </b></a><a NAME="E1:32_3_1_2_1_1_2_4_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_2_1_1_2_4_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:32_3_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2_1_1_2_4"><i><font color="Green">E63</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="E4:32_3_1_2_1_1_2_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E16:32_3_1_2_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R3" target="_self">is_Lout</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>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R4" target="_self">is_Rout</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>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:32"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan20.html#E2:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan20.html#E1:32_3_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan20.html#E12:32_3_1_2_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan20.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:32_3_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R3" target="_self">is_Lout</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>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R4" target="_self">is_Rout</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>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E41:32"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R3" target="_self">is_Lout</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>,<font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="jordan20.html#R4" target="_self">is_Rout</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>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/>


</div>
