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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <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/>



<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green">E39</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V3">constant</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="finseq_6.html#V1">circular</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V2">unfolded</a> )
 <b>and </b><br/><a NAME="E3:41"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="goboard5.html#V1">s.c.c.</a>
 <b>and </b><br/><a NAME="E4:41"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>and </b><br/><a NAME="E5:41"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="funct_1.html#V3">constant</a> <a href="finseq_6.html#V1">circular</a> <a href="topreal1.html#V1">special</a> <a href="topreal1.html#V2">unfolded</a> <a href="goboard5.html#V1">s.c.c.</a> <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>by </b><i><a class="txt" href="topreal8.html#E2:41"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topreal8.html#E3:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="topreal8.html#E4:41"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> non <a href="funct_1.html#V3">constant</a> <a href="finseq_6.html#V1">circular</a> <a href="topreal1.html#V1">special</a> <a href="topreal1.html#V2">unfolded</a> <a href="goboard5.html#V1">s.c.c.</a> <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="E8:41"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E2:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="topreal8.html#T3" target="_self">Th3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> 2;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> 1;<br/>
<a NAME="E9:41"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E8:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> 2 as  non <a href="realset1.html#V1">trivial</a> <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>by </b><i><a class="ref" href="realset1.html#T13">REALSET1:13</a></i>;<br/>
<a NAME="E11:41"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E8:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:41"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.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>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E13:41"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E14:41"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E8:41"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topreal8.html#E12:41"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E15:41"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> 1 as  non <a href="realset1.html#V1">trivial</a> <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>by </b><i><a class="ref" href="realset1.html#T13">REALSET1:13</a></i>;<br/>
<a NAME="E17:41"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="rfinseq.html#K1">/^</a> 1
 
<b>by </b><i><a class="ref" href="topreal8.html#T13" target="_self">Th13</a></i>;<br/>
<i><font color="Green">E50</font></i>: <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E17:41"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_6.html#T87">FINSEQ_6:87</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="graph_2.html#D2">GRAPH_2:def 2</a></i>
;<br/>
<a NAME="E19:41"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T20">FINSEQ_5:20</a></i>;<br/>
<a NAME="E20:41"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E53</font></i>: 
<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>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:41_2"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:41_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:41"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topreal8.html#E19:41"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E5:41_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:41_2"><i><font color="Green">E54</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:41_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:41_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topreal8.html#E4:41_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:41"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:41"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topreal8.html#E8:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal8.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E22:41"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="topreal1.html#V3">s.n.c.</a>
 
<b>by </b><i><a class="txt" href="topreal8.html#E8:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal8.html#T20" target="_self">Th20</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</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="E24:41"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E25:41"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="topreal1.html#V3">s.n.c.</a> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="topreal1.html#V1">special</a> )
 <b>and </b><br/><a NAME="E26:41"/><i><font color="Green">E56</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E27:41"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E28:41"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>7</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>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</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>9</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E29:41"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topreal8.html#E20:41"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topreal8.html#E21:41"><i><font color="Green">E53</font></i></a>, <a class="ref" href="goboard3.html#T1">GOBOARD3:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  non <a href="realset1.html#V1">trivial</a> <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>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topreal8.html#E29:41"><i><font color="Green">E59</font></i></a>, <a class="ref" href="realset1.html#T13">REALSET1:13</a></i>;<br/>
<a NAME="E31:41"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:41_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:41_3"/><b>then </b>
<font color="Maroon">c<sub>11</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>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T29">FINSEQ_5:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:41_3"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:41_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:41"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E6:41_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topreal8.html#E4:41_3"><i><font color="Green">E62</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:41_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:41_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topreal8.html#E5:41_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:41"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:41"><i><font color="Green">E42</font></i></a>, <a class="ref" href="topreal8.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E33:41"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="topreal1.html#V3">s.n.c.</a>
 
<b>by </b><i><a class="ref" href="topreal8.html#T21" target="_self">Th21</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</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="E35:41"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E36:41"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="topreal1.html#V3">s.n.c.</a> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="topreal1.html#V1">special</a> )
 <b>and </b><br/><a NAME="E37:41"/><i><font color="Green">E64</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E38:41"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E39:41"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>8</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>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</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>11</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E40:41"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K3">len</a> <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>11</sub></font>
 <b>by </b><i><a class="txt" href="topreal8.html#E31:41"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topreal8.html#E32:41"><i><font color="Green">E61</font></i></a>, <a class="ref" href="goboard3.html#T1">GOBOARD3:1</a></i>;<br/>
<a NAME="E41:41"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E15:41"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topreal8.html#E40:41"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as  non <a href="realset1.html#V1">trivial</a> <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>by </b><i><a class="ref" href="realset1.html#T13">REALSET1:13</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>12</sub></font>;<br/>

<a NAME="E43:41"/><i><font color="Green">E69</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E14:41"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E44:41"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E45:41"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E46:41"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>7</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>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<a NAME="E47:41"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>7</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>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E43:41"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>;<br/>
<a NAME="E48:41"/><i><font color="Green">E73</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E49:41"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>
<b>then </b><i><font color="Green">E75</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>2</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>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</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>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E13:41"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topreal8.html#E44:41"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
;<br/>
<b>thus </b><a NAME="E51:41"/>
<font color="Maroon">c<sub>13</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topreal8.html#E24:41"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topreal8.html#E35:41"><i><font color="Green">E62</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green">E72</font></i></a>, <a class="ref" href="topreal8.html#T12" target="_self">Th12</a></i>;<br/>

<a NAME="E52:41"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E41:41"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E53:41"/><b>then </b><i><font color="Green">E76</font></i>: 
<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>12</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E54:41"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topreal8.html#E29:41"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E55:41"/><b>then </b><i><font color="Green">E77</font></i>: 
<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>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E56:41"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:41"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topreal8.html#E12:41"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E57:41"/><b>then </b>
2 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E40:41"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E58:41"/><b>then </b><i><font color="Green">E78</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="sprect_3.html#T5">SPRECT_3:5</a></i>;<br/>
<a NAME="E59:41"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:41"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topreal8.html#E46:41"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topreal8.html#E49:41"><i><font color="Green">E74</font></i></a>, <a class="ref" href="topreal8.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E60:41"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E26:41"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topreal8.html#E27:41"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topreal8.html#E37:41"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topreal8.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E61:41"/>
1 <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>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topreal8.html#E29:41"><i><font color="Green">E59</font></i></a>, <a class="ref" href="sprect_3.html#T8">SPRECT_3:8</a></i>;<br/>
<a NAME="E62:41"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>10</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>10</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>10</sub></font>,<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>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E55:41"><i><font color="Green">E77</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<a NAME="E63:41"/>
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,1
 
<b>by </b><i><a class="txt" href="topreal8.html#E41:41"><i><font color="Green">E68</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<a NAME="E64:41"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E28:41"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topreal8.html#E62:41"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E65:41"/><i><font color="Green">E82</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,1 <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,<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>12</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E36:41"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topreal8.html#E58:41"><i><font color="Green">E78</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/>
<a NAME="E66:41"/>
1 <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>12</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E58:41"><i><font color="Green">E78</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E67:41"/><b>then </b>
<font color="Maroon">c<sub>12</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>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,<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>12</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E53:41"><i><font color="Green">E76</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<a NAME="E68:41"/><b>then </b>
not <font color="Maroon">c<sub>12</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>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,1
 
<b>by </b><i><a class="txt" href="topreal8.html#E65:41"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E69:41"/><b>then </b><i><font color="Green">E83</font></i>: 
not <font color="Maroon">c<sub>12</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>12</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E70:41"/><i><font color="Green">E84</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>10</sub></font>,<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>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/>
<a NAME="E71:41"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,1 <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/>
<a NAME="E72:41"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E60:41"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topreal8.html#E70:41"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E73:41"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>12</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E27:41"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topreal8.html#E39:41"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topreal8.html#E50:41"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topreal8.html#E64:41"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topreal8.html#E69:41"><i><font color="Green">E83</font></i></a>, <a class="ref" href="topreal8.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E74:41"/>
<font color="Maroon">c<sub>13</sub></font> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i><a class="txt" href="topreal8.html#E25:41"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topreal8.html#E36:41"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green">E72</font></i></a>, <a class="ref" href="topreal8.html#T34" target="_self">Th34</a></i>;<br/>

<b>thus </b><a NAME="E75:41"/>
<font color="Maroon">c<sub>13</sub></font> is <a href="goboard5.html#V1">s.c.c.</a>
 <b>by </b><i><a class="txt" href="topreal8.html#E25:41"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topreal8.html#E27:41"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topreal8.html#E36:41"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topreal8.html#E39:41"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topreal8.html#E50:41"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topreal8.html#E60:41"><i><font color="Green">E79</font></i></a>, <a class="ref" href="topreal8.html#T33" target="_self">Th33</a></i>;<br/>

<b>thus </b><a NAME="E76:41"/>
<font color="Maroon">c<sub>13</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="topreal8.html#E25:41"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topreal8.html#E36:41"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green">E72</font></i></a>, <a class="ref" href="topreal8.html#T26" target="_self">Th26</a></i>;<br/>

<b>thus </b> <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E18:41"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green">E72</font></i></a>, <a class="ref" href="topreal8.html#T35" target="_self">Th35</a></i>
<br/>.= 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="topreal8.html#E26:41"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topreal8.html#E37:41"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green">E72</font></i></a>, <a class="ref" href="topreal8.html#T35" target="_self">Th35</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="topreal8.html#E48:41"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="topreal8.html#E27:41"><i><font color="Green">E57</font></i></a>, <a class="ref" href="graph_2.html#T57">GRAPH_2:57</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>2</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>2</sub></font></span>)</span> = 
<font color="Maroon">c<sub>8</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>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E13:41"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topreal8.html#E44:41"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</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>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E39:41"><i><font color="Green">E66</font></i></a>, <a class="ref" href="graph_2.html#T58">GRAPH_2:58</a></i>
;<br/>

<a NAME="E80:41"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E18:41"><i><font color="Green">E50</font></i></a>, <a class="ref" href="graph_2.html#T13">GRAPH_2:13</a></i>;<br/>
<a NAME="E81:41"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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>10</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_2.html#T13">GRAPH_2:13</a></i>;<br/>
<a NAME="E82:41"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E29:41"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topreal8.html#E40:41"><i><font color="Green">E67</font></i></a>, <a class="txt" href="topreal8.html#E80:41"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E83:41"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>


</div>
