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

<b>let </b><font color="Maroon">d</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">l</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">r</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">r'</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">l'</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>;<br/>









<b>assume </b><a NAME="E1:47"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font>
 ;<br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font>
 ;<br/>

<b>thus </b><a NAME="E3:47"/>
(  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l'</font>,<font color="Maroon">r'</font> implies  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> st <br/>( <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E52</font></i>: 
 <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l'</font>,<font color="Maroon">r'</font>
 ;<br/>

<b>assume </b><a NAME="E2:47_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</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">d</font>,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( <font color="Maroon">l</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">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">r'</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">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E3:47_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  ex <font color="Olive">xi</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">xi</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1_1"/>
( ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) or ( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) or <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_1"><b>suppose </b></a><a NAME="E1:47_1_1_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 ;<br/><div class="add"><a NAME="E2:47_1_1_1_1"/>
<font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">xi</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:47_1_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">xi</font> &amp; <font color="Maroon">xi</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">xi</font>
;<br/><b>thus </b><a NAME="E5:47_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">xi</font>]
 <b>by </b><i>, , <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="47_1_1_1_2"><b>suppose </b></a><a NAME="E1:47_1_1_1_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
;<br/><b>thus </b><a NAME="E2:47_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>]
 <b>by </b><i>, <a class="txt" href="chain_1.html#E1:47"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_3"><b>suppose </b></a><a NAME="E1:47_1_1_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
;<br/><b>thus </b><a NAME="E2:47_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>]
 <b>by </b><i>, <a class="txt" href="chain_1.html#E1:47"><i><font color="Green">E37</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">x</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:47_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="chain_1.html#T2" target="_self">Th2</a>);<br/></i>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> <b>by </b><i/>;<br/>
<a NAME="E7:47_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>
<a NAME="E9:47_1"/>
<font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:47_1"/><b>then </b>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> st <br/>( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; ( <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) )
 
<b>by </b><i><a class="ref" href="chain_1.html#T1" target="_self">Th1</a>, , </i>;<br/>
<b>hence </b><a NAME="E11:47_1"/>
contradiction
 <b>by </b><i/>;<br/>


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

<b>given </b><font color="Maroon">i0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font><b> such that </b><a NAME="E4:47"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> or <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> )
 ;<br/>

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

<b>assume </b><a NAME="E5:47"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> ;<br/>
<a NAME="E7:47"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> )
 
<b>by </b><i>, <a class="txt" href="chain_1.html#E1:47"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<a NAME="E8:47"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> st <br/>( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; ( <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">i0</font>
;<br/>

<b>thus </b><a NAME="E1:47_2"/>
( <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> or <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> ) )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T1" target="_self">Th1</a>, <a class="ref" href="chain_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:47"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l'</font>,<font color="Maroon">r'</font>
 ;<br/>


</div>
