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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</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>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="chain_1.html#M1" target="_self">Grating</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>















<b>assume </b><a NAME="E1:64"/><i><font color="Green">E48</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> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp;  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:64"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>assume </b><a NAME="E3:64"/><i><font color="Green">E50</font></i>: 
 <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:64"/><i><font color="Green">E51</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<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>3</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</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>4</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>5</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>8</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) or ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</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>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</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#E1:64"><i><font color="Green">E48</font></i></a>, <a class="txt" href="chain_1.html#E2:64"><i><font color="Green">E49</font></i></a>, <a class="ref" href="chain_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E6:64"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="chain_1.html#E2:64"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="chain_1.html#E5:64"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>
;<br/>
<a NAME="E8:64"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="chain_1.html#E6:64"><i><font color="Green">E52</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:64"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="chain_1.html#E10:64"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E12:64"/>
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E10:64"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E13:64"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</sub></font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E5:64"><i><font color="Green">E51</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_2"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</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>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</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>11</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>11</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> ) or <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</sub></font> )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E1:64"><i><font color="Green">E48</font></i></a>, <a class="txt" href="chain_1.html#E2:64"><i><font color="Green">E49</font></i></a>, <a class="txt" href="chain_1.html#E3:64"><i><font color="Green">E50</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="64_2_1"><b>suppose </b></a><a NAME="E1:64_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</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>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_2_1"/>
ex <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>3</sub></font> st <br/>( ( <font color="Maroon">c<sub>4</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> &amp; <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>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) or ( <font color="Maroon">c<sub>4</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>7</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>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>7</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#E13:64"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_2"><b>suppose </b></a><a NAME="E1:64_2_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</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>11</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>11</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_2_2"/>
ex <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>3</sub></font> st <br/>( ( <font color="Maroon">c<sub>4</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> &amp; <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>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) or ( <font color="Maroon">c<sub>4</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>7</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>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>7</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="64_2_3"><b>suppose </b></a><a NAME="E1:64_2_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_2_3"/>
ex <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>3</sub></font> st <br/>( ( <font color="Maroon">c<sub>4</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> &amp; <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>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) or ( <font color="Maroon">c<sub>4</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>7</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>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>7</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="64_2_4"><b>suppose </b></a><a NAME="E1:64_2_4"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</sub></font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:64_2_4"/><i><font color="Green">E56</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>3</sub></font> holds <br/>( ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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 <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#NR2" target="_self">&lt;&gt;</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; ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <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> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> or <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#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><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>3</sub></font>,  <a href="real_1.html#NM1">Real</a>] means ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</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">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</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">a<sub>1</sub></font> );<br/><a NAME="E3:64_2_4"/><i><font color="Green">E57</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>3</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="64_2_4_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>3</sub></font>;<br/>

<a NAME="E1:64_2_4_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>12</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>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</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>12</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>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</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>8</sub></font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:64"><i><font color="Green">E48</font></i></a>, <a class="txt" href="chain_1.html#E2:64"><i><font color="Green">E49</font></i></a>, <a class="txt" href="chain_1.html#E1:64_2_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="chain_1.html#T35" target="_self">Th35</a>, <a class="ref" href="chain_1.html#T36" target="_self">Th36</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_2_4_1_1"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</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>12</sub></font> ) or <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:64_2_4_1_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</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>12</sub></font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
;<br/><b>thus </b><a NAME="E2:64_2_4_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>]
 <b>by </b><i><a class="txt" href="chain_1.html#E5:64"><i><font color="Green">E51</font></i></a>, <a class="txt" href="chain_1.html#E1:64_2_4_1_1_1"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:64_2_4_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:64_2_4_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:64_2_4_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E3:64_2_4_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:64_2_4_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:64_2_4_1_1_2"/><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>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E2:64_2_4"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E5:64_2_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</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>12</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E5:64"><i><font color="Green">E51</font></i></a>, <a class="txt" href="chain_1.html#E3:64_2_4_1_1_2"><i><font color="Green">E61</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:64_2_4_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</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>12</sub></font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T1" target="_self">Th1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/><a NAME="E8:64_2_4_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</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>12</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:64_2_4_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E9:64_2_4_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>]
 <b>by </b><i><a class="txt" href="chain_1.html#E2:64_2_4_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="chain_1.html#E7:64_2_4_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_4_1_1_3"><b>suppose </b></a><a NAME="E1:64_2_4_1_1_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:64_2_4_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:64_2_4_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E3:64_2_4_1_1_3"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:64_2_4_1_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:64_2_4_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:64_2_4_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E5:64_2_4_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>12</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>12</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E5:64"><i><font color="Green">E51</font></i></a>, <a class="txt" href="chain_1.html#E3:64_2_4_1_1_3"><i><font color="Green">E60</font></i></a></i>;<br/><b>hence </b><a NAME="E6:64_2_4_1_1_3"/>
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>]
 <b>by </b><i><a class="txt" href="chain_1.html#E2:64_2_4"><i><font color="Green">E56</font></i></a>, <a class="txt" href="chain_1.html#E3:64_2_4_1_1_3"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</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>3</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:64_2_4"/><i><font color="Green">E58</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>3</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#E3:64_2_4"><i><font color="Green">E57</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>3</sub></font> <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E7:64_2_4"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E5:64_2_4"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E8:64_2_4"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>11</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>11</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>3</sub></font> st <font color="Maroon">c<sub>7</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> holds <br/>( <font color="Maroon">c<sub>7</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>13</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="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> ) ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:64_2_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="chain_1.html#E5:64_2_4"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E9:64_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="chain_1.html#E3:64"><i><font color="Green">E50</font></i></a>, <a class="txt" href="chain_1.html#E7:64_2_4"><i><font color="Green">E59</font></i></a>, <a class="ref" href="chain_1.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
