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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="jordan23.html#V3" target="_self">poorly-one-to-one</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V3">s.n.c.</a> )
 ;<br/>

<b>assume </b><a NAME="E2:40"/><i><font color="Green">E20</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E3:40"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:40"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan23.html#E2:40"><i><font color="Green">E20</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<a NAME="E6:40"/><i><font color="Green">E23</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:40"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan23.html#E5:40"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>assume </b><a NAME="E7:40"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E8:40"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E3:40"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E9:40"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:40"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan23.html#E5:40"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan23.html#E6:40"><i><font color="Green">E23</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/>
<a NAME="E10:40"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T40">JORDAN3:40</a></i>;<br/>
<b>assume </b><a NAME="E11:40"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E12:40"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E5:40"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan23.html#E10:40"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E13:40"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E14:40"/><i><font color="Green">E28</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26">TOPREAL3:26</a></i>;<br/>
<a NAME="E15:40"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E9:40"><i><font color="Green">E25</font></i></a>, <a class="ref" href="jordan3.html#T42">JORDAN3:42</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
( <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan23.html#E13:40"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_1_1"/><i><font color="Green">E31</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan23.html#E2:40"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan23.html#E5:40"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan23.html#E9:40"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan23.html#E14:40"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan23.html#E1:40_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/><a NAME="E3:40_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="txt" href="jordan23.html#E1:40"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan23.html#E1:40_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E4:40_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i><a class="txt" href="jordan23.html#E9:40"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan23.html#E15:40"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:40_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan23.html#E2:40"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan23.html#E5:40"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:40_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan23.html#E6:40"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="jordan23.html#E7:40"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan23.html#E4:40_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/><a NAME="E8:40_1_1"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="jordan23.html#E2:40_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E9:40_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan23.html#E2:40"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan23.html#E5:40"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan23.html#E1:40_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:40_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E11:40_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E1:40"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan23.html#E5:40"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan23.html#E6:40"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan23.html#E5:40_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan23.html#E7:40_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan23.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan23.html#E9:40"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan23.html#E15:40"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:40_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><b>hence </b><a NAME="E4:40_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan23.html#E1:40"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan23.html#E1:40_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topreal1.html#D9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
