<?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="E1:63"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</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/>

<b>assume </b><a NAME="E2:63"/><i><font color="Green">E48</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>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>3</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_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>3</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>7</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>6</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>7</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 ( <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>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>6</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> ) ) 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>3</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>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>6</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>7</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> ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E1:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="chain_1.html#T35" target="_self">Th35</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>suppose </b></a><a NAME="E1:63_1_1"/><i><font color="Green">E49</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>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>7</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>6</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>7</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 ( <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>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>6</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> ) )
 ;<br/><div class="add"><a NAME="E2:63_1_1"/><b>then </b>
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">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>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/><a NAME="E3:63_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</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>9</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63"><i><font color="Green">E48</font></i></a>, <a class="ref" href="chain_1.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E4:63_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E5:63_1_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_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>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E1:63_1_1"><i><font color="Green">E49</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_1_1_1_1"/><i><font color="Green">E52</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>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</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="Maroon">c<sub>9</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>9</sub></font> )
 ;<br/><a NAME="E2:63_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:63_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="chain_1.html#E4:63_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:63_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="chain_1.html#T36" target="_self">Th36</a></i>;<br/><b>hence </b><a NAME="E4:63_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="chain_1.html#E2:63_1_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</sub></font> )
 ;<br/><a NAME="E2:63_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:63_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="chain_1.html#E4:63_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:63_1_1_1_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>9</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>9</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="chain_1.html#T36" target="_self">Th36</a></i>;<br/><b>hence </b><a NAME="E4:63_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="chain_1.html#E2:63_1_1_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:63_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:63_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="chain_1.html#E2:63_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:63_1_1_1_1_2"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:63_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="chain_1.html#E4:63_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</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="63_1_2"><b>suppose </b></a><a NAME="E1:63_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>3</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>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>6</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>7</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> )
 ;<br/><div class="add"><a NAME="E2:63_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</sub></font> )
 ;<br/><b>thus </b><a NAME="E3:63_1_2"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_2_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>3</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>4</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 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="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 ( <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> ) ) )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E1:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="chain_1.html#T35" target="_self">Th35</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_2_1_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>3</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>4</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> )
 ;<br/><div class="add"><a NAME="E2:63_1_2_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</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>9</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63"><i><font color="Green">E48</font></i></a>, <a class="ref" href="chain_1.html#T30" target="_self">Th30</a></i>;<br/><div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:63_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63_1_2_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:63_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="chain_1.html#T36" target="_self">Th36</a></i>;<br/><b>hence </b><a NAME="E4:63_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="chain_1.html#E2:63_1_2_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1_2_1_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>9</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>9</sub></font>
 ;<br/><a NAME="E2:63_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63_1_2_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:63_1_2_1_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>9</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>9</sub></font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="chain_1.html#T36" target="_self">Th36</a></i>;<br/><b>hence </b><a NAME="E4:63_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="chain_1.html#E2:63_1_2_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:63_1_2_1_1_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:63_1_2_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_2_1_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>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="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 ( <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> ) )
 ;<br/><div class="add"><a NAME="E2:63_1_2_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>9</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>9</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>9</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>9</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>9</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>9</sub></font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63"><i><font color="Green">E47</font></i></a>, <a class="ref" href="chain_1.html#T36" target="_self">Th36</a></i>;<br/><b>hence </b><a NAME="E3:63_1_2_1_1_2"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</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>9</sub></font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self">Th17</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>
