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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="chain_1.html#M1" target="_self">Grating</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:84"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:84"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:84"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84"><i><font color="Green">E57</font></i></a>, <a class="ref" href="chain_1.html#T42" target="_self">Th42</a></i>;<br/>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><b>hence </b><a NAME="E2:84_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:84"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<a NAME="E1:84_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( ( <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> ) ) ) or for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:84_2_1"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="ref" href="chain_1.html#T20" target="_self">Th20</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E2:84_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>,  <a href="real_1.html#NM1">Real</a>] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/><a NAME="E2:84_2_1_1_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:84_2_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E2:84_2_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:84_2_1_1_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E2:84_2_1_1_1"><i><font color="Green">E63</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/><b>thus </b><a NAME="E6:84_2_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( ( <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="chain_1.html#E4:84_2_1_1_1"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:84_2_1_1_2"/><i><font color="Green">E63</font></i>: 
(  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#D11" target="_self">Def11</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/><a NAME="E4:84_2_1_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:84_2_1_1_2"><i><font color="Green">E63</font></i></a></i>;<br/><b>hence </b><a NAME="E5:84_2_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( ( <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:84_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="chain_1.html#E3:84_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="chain_1.html#T23" target="_self">Th23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:84_2"/><i><font color="Green">E61</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( ( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 ;<br/>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E2:84_2_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:84_2_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_2"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:84_2"/><i><font color="Green">E63</font></i>: 
( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_3_1"/>
( ( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_3_1_1"><b>suppose </b></a><a NAME="E1:84_2_3_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_3_1_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E2:84_2_3_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:84_2_3_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_3_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_3_1_1_1"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:84_2_3_1_1"/>
( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_3_1_2"><b>suppose </b></a><a NAME="E1:84_2_3_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_3_1_2"/>
( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="chain_1.html#E4:84_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="chain_1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E7:84_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( ( <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> ) ) ) or for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_4"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:84_2_4"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="ref" href="chain_1.html#T19" target="_self">Th19</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_4_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E2:84_2_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_4_1_1"><b>suppose </b></a><a NAME="E1:84_2_4_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>,  <a href="real_1.html#NM1">Real</a>] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/><a NAME="E2:84_2_4_1_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:84_2_4_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E2:84_2_4_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:84_2_4_1_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E2:84_2_4_1_1"><i><font color="Green">E66</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/><b>thus </b><a NAME="E6:84_2_4_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:84_2_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_4_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="chain_1.html#E4:84_2_4_1_1"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_4_1_2"><b>suppose </b></a><a NAME="E1:84_2_4_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:84_2_4_1_2"/><i><font color="Green">E66</font></i>: 
(  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#D11" target="_self">Def11</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/><a NAME="E4:84_2_4_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:84_2_4_1_2"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E5:84_2_4_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( ( <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:84_2_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_4_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="chain_1.html#E3:84_2_4_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="chain_1.html#T23" target="_self">Th23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:84_2"/><i><font color="Green">E64</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( ( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 ;<br/>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_5"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E2:84_2_5"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:84_2_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_5"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:84_2"/>
( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_6_1"/>
( ( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_6_1_1"><b>suppose </b></a><a NAME="E1:84_2_6_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_6_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_6_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> implies <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E2:84_2_6_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:84_2_6_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_6_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_6_1_1_1"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:84_2_6_1_1"/>
( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_6_1_2"><b>suppose </b></a><a NAME="E1:84_2_6_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_6_1_2"/>
( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="chain_1.html#E10:84_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="chain_1.html#T41" target="_self">Th41</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

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

<a NAME="E13:84_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_7_1"/>
( ( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_7_1_1"><b>suppose </b></a><a NAME="E1:84_2_7_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 ;<br/><div class="add"><div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_7_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_7_1_1"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E2:84_2_7_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_7_1_1"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_7_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_7_1_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_7_1_1"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E2:84_2_7_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_7_1_1"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:84_2_7_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_7_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_7_1_2"><b>suppose </b></a><a NAME="E1:84_2_7_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_7_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E5:84"><i><font color="Green">E60</font></i></a>, <a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="chain_1.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E14:84_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84"><i><font color="Green">E57</font></i></a></i>;<br/>

<a NAME="E15:84_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_8_1"/>
( ( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_8_1_1"><b>suppose </b></a><a NAME="E1:84_2_8_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 ;<br/><div class="add"><div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_8_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_8_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_8_1_1"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E2:84_2_8_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_8_1_1"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_8_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_8_1_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_8_1_1"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E2:84_2_8_1_1_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_8_1_1"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:84_2_8_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_8_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_8_1_2"><b>suppose </b></a><a NAME="E1:84_2_8_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_8_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E5:84"><i><font color="Green">E60</font></i></a>, <a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="chain_1.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E16:84_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84"><i><font color="Green">E57</font></i></a></i>;<br/>

<a NAME="E17:84_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E18:84_2"/>
( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>9</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E5:84_2"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:84_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E17:84_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="chain_1.html#T32" target="_self">Th32</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E20:84_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="chain_1.html#E1:84"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E23:84_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E24:84_2"/><i><font color="Green">E69</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">c<sub>3</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E23:84_2"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E25:84_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E20:84_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="chain_1.html#T51" target="_self">Th51</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_9"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_9"/>
( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E23:84_2"><i><font color="Green">E68</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_9_1"><b>suppose </b></a><a NAME="E1:84_2_9_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:84_2_9_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:84_2_9_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_9_1"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E3:84_2_9_1_1"/><b>then </b>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:84"><i><font color="Green">E58</font></i></a>, <a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E23:84_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="chain_1.html#E25:84_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="chain_1.html#T46" target="_self">Th46</a></i>;<br/><b>hence </b><a NAME="E4:84_2_9_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_1"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div><b>thus </b><a NAME="E3:84_2_9_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_9_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:84_2_9_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_9_1"><i><font color="Green">E71</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_9_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_9_1_2_1"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E2:84"><i><font color="Green">E58</font></i></a>, <a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E23:84_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="chain_1.html#E25:84_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="chain_1.html#T46" target="_self">Th46</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_9_1_2_1_1"><b>suppose </b></a><a NAME="E1:84_2_9_1_2_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><a NAME="E2:84_2_9_1_2_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E24:84_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="chain_1.html#T22" target="_self">Th22</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_9_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_9_1_2_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_9_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_2_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_9_1_2_1_1"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E2:84_2_9_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or ( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_9_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>hence </b><a NAME="E3:84_2_9_1_2_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E24:84_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_2_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="chain_1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E4:84_2_9_1_2_1_1_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or ( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_9_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_2_1_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><b>hence </b><a NAME="E5:84_2_9_1_2_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_2_1_1"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:84_2_9_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E6:84_2_9_1_2_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E23:84_2"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_9_1_2_1_2"><b>suppose </b></a><a NAME="E1:84_2_9_1_2_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><a NAME="E2:84_2_9_1_2_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E24:84_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="chain_1.html#T21" target="_self">Th21</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_9_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:84_2_9_1_2_1_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_2_9_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_2_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_9_1_2_1_2"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E2:84_2_9_1_2_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or ( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_9_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>hence </b><a NAME="E3:84_2_9_1_2_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E24:84_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_2_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="chain_1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E4:84_2_9_1_2_1_2_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or ( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E2:84_2_9_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_2_1_2_1"><i><font color="Green">E76</font></i></a></i>;<br/><b>hence </b><a NAME="E5:84_2_9_1_2_1_2_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_1_2_1_2"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:84_2_9_1_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E6:84_2_9_1_2_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E23:84_2"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_9_2"><b>suppose </b></a><a NAME="E1:84_2_9_2"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:84_2_9_2"/><i><font color="Green">E72</font></i>: 
( ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:84"><i><font color="Green">E58</font></i></a>, <a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E23:84_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="chain_1.html#E25:84_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="chain_1.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E4:84_2_9_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E4:84"><i><font color="Green">E59</font></i></a>, <a class="txt" href="chain_1.html#E3:84_2_9_2"><i><font color="Green">E72</font></i></a></i>;<br/><b>thus </b><a NAME="E5:84_2_9_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_9_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_9_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_9_2_1_1"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E3:84_2_9_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="chain_1.html#E4:84_2_9_2"><i><font color="Green">E73</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_9_2_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_9_2_1_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><a NAME="E2:84_2_9_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E24:84_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="chain_1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E3:84_2_9_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:84_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="chain_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E4:84_2_9_2_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E23:84_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="chain_1.html#T49" target="_self">Th49</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_9_2_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_9_2_1_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><a NAME="E2:84_2_9_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E24:84_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="chain_1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E3:84_2_9_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E9:84_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="chain_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E4:84_2_9_2_1_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E23:84_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="chain_1.html#E1:84_2_9_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="chain_1.html#T49" target="_self">Th49</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E7:84"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="chain_1.html#T6" target="_self">Th6</a></i>;<br/>


</div>
