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

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p3</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>







<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E1:17"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="finseq_4.html#K3">*&gt;</a></span>
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E57</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a>, <a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>;<br/>
<a NAME="E3:17"/><i><font color="Green">E60</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:17"/><i><font color="Green">E63</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:17_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<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="E1:17_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:17_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>
<a NAME="E4:17_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<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="E2:17_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_1"><b>suppose </b></a><a NAME="E1:17_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:17_1_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,1
 <b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:17_1_2_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:17_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2_2"><b>suppose </b></a><a NAME="E1:17_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font>
 ;<br/><div class="add"><a NAME="E2:17_1_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,2
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:17_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E5:17"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>


</div>
