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

<b>set </b><font color="Maroon">L1</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">L2</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">L3</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">L4</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">p0</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">p01</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">p10</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">p1</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E1:41"/><i><font color="Green">E37</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E2:41"/><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> as   <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>reconsider </b><font color="Maroon">f2</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> as   <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>take </b>
<font color="Maroon">f1</font>
;<br/>

<b>take </b>
<font color="Maroon">f2</font>
;<br/>

<a NAME="E6:41"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E7:41"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>;<br/>
<b>thus </b><a NAME="E8:41"/>
<font color="Maroon">f1</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_1"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>hence </b><a NAME="E2:41_1"/>
<font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_3.html#T104">FINSEQ_3:104</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D10" target="_self">TOPREAL1:def 10</a><br/>

<b>thus </b><a NAME="E3:41_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E4:41_1"/>
<font color="Maroon">f1</font> is <a href="topreal1.html#V2" target="_self">unfolded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D8" target="_self">TOPREAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E56</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> )
 ;<br/>

<a NAME="E2:41_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:41_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E4:41_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:41_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
(  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,1 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> &amp;  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:41_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> iff <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_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> ;<br/>

<b>thus </b><a NAME="E1:41_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/>

<a NAME="E2:41_1_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:41_1_1_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 ex <font color="Olive">p</font> being  <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> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
;<br/>
<a NAME="E4:41_1_1_1_1"/>
 ex <font color="Olive">p2</font> being  <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> st <br/>( <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:41_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>


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

<b>assume </b><a NAME="E2:41_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>
 ;<br/>

<a NAME="E3:41_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:41_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:41_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E5:41_1"/>
<font color="Maroon">f1</font> is <a href="topreal1.html#V3" target="_self">s.n.c.</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D9" target="_self">TOPREAL1:def 9</a><br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:41_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_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="E1:41_1_2_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not <font color="Maroon">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">f1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:41_1_2_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_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="E1:41_1_2_1_1_1_1_1"/>
( ( 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">f1</font> ) or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> or  not <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">f1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:41_1_2_1_1_1_1_1_1"/>
( 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">f1</font> )
 ;<br/><div class="add"><a NAME="E2:41_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:41_1_2_1_1_1_1_1_1"/>
contradiction
 <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="41_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:41_1_2_1_1_1_1_1_2"/>
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> or  not <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">f1</font> )
 ;<br/><div class="add"><a NAME="E2:41_1_2_1_1_1_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:41_1_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:41_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_2_1_1_2"/>
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not <font color="Maroon">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">f1</font> )
 ;<br/><div class="add"><a NAME="E2:41_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:41_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:41_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>


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

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7" target="_self">TOPREAL1:def 7</a><br/>

<b>assume </b><a NAME="E6:41_1"/><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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">f1</font> )
 ;<br/>

<a NAME="E7:41_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<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="41_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_3"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, , <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_1"><b>suppose </b></a><a NAME="E1:41_1_3_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:41_1_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_3_2"><b>suppose </b></a><a NAME="E1:41_1_3_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:41_1_3_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<a NAME="E9:41"/><i><font color="Green">E77</font></i>: 
 <a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<a NAME="E1:41_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:41_2"/><b>then </b>
( 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">f1</font> &amp;  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,1 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:41_2"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</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">f1</font> ) </span>}</span> 
 
;<br/>
<a NAME="E4:41_2"/>
( 2 <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">f1</font> &amp;  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,2 )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:41_2"/><b>then </b>
 <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</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">f1</font> ) </span>}</span> 
 
;<br/>
<a NAME="E6:41_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</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">f1</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E7:41_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</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">f1</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" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:41_2_1"/>
<font color="Maroon">a</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" target="_self">LSeg</a> <font color="Maroon">f1</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">f1</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</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="E3:41_2_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f1</font>,<font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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">f1</font> )
 ;<br/>
<a NAME="E4:41_2_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E5:41_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:41_2_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<a NAME="E7:41_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E8:41_2_1"/>
<font color="Maroon">a</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" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></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>
<b>hence </b><a NAME="E8:41_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:41"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>
<a NAME="E11:41"/><i><font color="Green">E82</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E12:41"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>;<br/>
<b>thus </b><a NAME="E13:41"/>
<font color="Maroon">f2</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:41_3"/>
<font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T104">FINSEQ_3:104</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D10" target="_self">TOPREAL1:def 10</a><br/>

<b>thus </b><a NAME="E2:41_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E3:41_3"/>
<font color="Maroon">f2</font> is <a href="topreal1.html#V2" target="_self">unfolded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D8" target="_self">TOPREAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:41_3_1"/><i><font color="Green">E104</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 ;<br/>

<a NAME="E2:41_3_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:41_3_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E4:41_3_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:41_3_1"/><b>then </b><i><font color="Green">E108</font></i>: 
(  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,1 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> &amp;  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:41_3_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> iff <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_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> ;<br/>

<b>thus </b><a NAME="E1:41_3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/>

<a NAME="E2:41_3_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p2</font> where <font color="Olive">p2</font> is   <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> : ( <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:41_3_1_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
 ex <font color="Olive">p</font> being  <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> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
;<br/>
<a NAME="E4:41_3_1_1_1"/>
 ex <font color="Olive">p2</font> being  <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> st <br/>( <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="topreal1.html#E3:41"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:41_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>


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

<b>assume </b><a NAME="E2:41_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 ;<br/>

<a NAME="E3:41_3_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:41_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:41_3_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="txt" href="topreal1.html#E1:41"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topreal1.html#E2:41"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E4:41_3"/>
<font color="Maroon">f2</font> is <a href="topreal1.html#V3" target="_self">s.n.c.</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D9" target="_self">TOPREAL1:def 9</a><br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:41_3_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_3_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not <font color="Maroon">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">f2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_2_1_1"><b>suppose </b></a><a NAME="E1:41_3_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:41_3_2_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_2_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="E1:41_3_2_1_1_1_1"/>
( ( 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">f2</font> ) or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> or  not <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">f2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:41_3_2_1_1_1_1_1"/>
( 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">f2</font> )
 ;<br/><div class="add"><a NAME="E2:41_3_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:41_3_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="topreal1.html#E1:41"><i><font color="Green">E37</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="41_3_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:41_3_2_1_1_1_1_2"/>
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> or  not <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">f2</font> )
 ;<br/><div class="add"><a NAME="E2:41_3_2_1_1_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:41_3_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:41_3_2_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_2_1_2"><b>suppose </b></a><a NAME="E1:41_3_2_1_2"/>
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not <font color="Maroon">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">f2</font> )
 ;<br/><div class="add"><a NAME="E2:41_3_2_1_2"/><b>then </b>
 <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:41_3_2_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7" target="_self">TOPREAL1:def 7</a><br/>

<b>assume </b><a NAME="E5:41_3"/><i><font color="Green">E124</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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">f2</font> )
 ;<br/>

<a NAME="E6:41_3"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<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="41_3_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_3_3"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, <a class="txt" href="topreal1.html#E1:41"><i><font color="Green">E37</font></i></a>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_3_1"><b>suppose </b></a><a NAME="E1:41_3_3_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="txt" href="topreal1.html#E2:41"><i><font color="Green">E38</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:41_3_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_3_2"><b>suppose </b></a><a NAME="E1:41_3_3_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, <a class="txt" href="topreal1.html#E2:41"><i><font color="Green">E38</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:41_3_3_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<a NAME="E14:41"/><i><font color="Green">E159</font></i>: 
 <a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>proof </b></a><div class="add">

<a NAME="E1:41_4"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:41_4"/><b>then </b>
( 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">f2</font> &amp;  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,1 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:41_4"/><b>then </b><i><font color="Green">E160</font></i>: 
 <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</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">f2</font> ) </span>}</span> 
 
;<br/>
<a NAME="E4:41_4"/>
( 2 <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">f2</font> &amp;  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,2 )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:41_4"/><b>then </b>
 <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</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">f2</font> ) </span>}</span> 
 
;<br/>
<a NAME="E6:41_4"/><b>then </b><i><font color="Green">E161</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</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">f2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topreal1.html#E1:41"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E7:41_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</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">f2</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" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:41_4_1"/>
<font color="Maroon">a</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" target="_self">LSeg</a> <font color="Maroon">f2</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">f2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</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="E3:41_4_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4" target="_self">LSeg</a> <font color="Maroon">f2</font>,<font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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">f2</font> )
 ;<br/>
<a NAME="E4:41_4_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal1.html#E3:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E5:41_4_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:41_4_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="txt" href="topreal1.html#E3:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<a NAME="E7:41_4_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, <a class="txt" href="topreal1.html#E3:41"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E8:41_4_1"/>
<font color="Maroon">a</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" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></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>
<b>hence </b><a NAME="E8:41_4"/>
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E2:41"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:41"/><b>then </b><i><font color="Green">E163</font></i>: 
 <a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>
<b>thus </b><a NAME="E16:41"/>
 <a href="topreal1.html#K2" target="_self">R^2-unit_square</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_5"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_5"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:41_5"/><i><font color="Green">E165</font></i>: 
 ex <font color="Olive">p</font> being  <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> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="topreal1.html#E2:41"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E5:41_5"/>
 ex <font color="Olive">p2</font> being  <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> st <br/>( <font color="Olive">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p2</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="topreal1.html#E2:41"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E6:41_5"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal1.html#E3:41"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:41"/><b>then </b><i><font color="Green">E166</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E19:41"/><i><font color="Green">E167</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" target="_self">L~</a> <font color="Maroon">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> </span>)</span>
<b>by </b><i>, <a class="txt" href="topreal1.html#E1:41"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> </span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 ) </span>}</span> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p3">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : ( <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="txt" href="topreal1.html#E2:41"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topreal1.html#E3:41"><i><font color="Green">E39</font></i></a>, , , , <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
;<br/>

<b>thus </b><a NAME="E21:41"/>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>;<br/>


</div>
