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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finseq_6.html#V1">circular</a> <a href="goboard5.html#V1">s.c.c.</a> <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="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E25</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E26</font></i>: 
 <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> 4
 ;<br/>

<a NAME="E3:27"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_1"/><i><font color="Green">E27</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:27_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:27_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E4:27_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:27_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:27_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E6:27_1"/><b>then </b><i><font color="Green">E30</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E7:27_1"/>
0 <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="txt" href="topreal8.html#E2:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:27_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E9:27_1"/><b>then </b><i><font color="Green">E31</font></i>: 
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E10:27_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:27_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E11:27_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</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="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E12:27_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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E2:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:27_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E11:27_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E14:27_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E15:27_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</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="topreal8.html#E12:27_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E16:27_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:27_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:27_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E11:27_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E18:27_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E19:27_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</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="topreal8.html#E16:27_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E20:27_1"/><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:27_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="topreal8.html#E2:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>;<br/>
<b>assume </b><a NAME="E21:27_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="topreal8.html#E21:27_1"><i><font color="Green">E38</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="27_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal8.html#E2:27"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topreal8.html#E3:27_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topreal8.html#E6:27_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topreal8.html#E15:27_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topreal8.html#E20:27_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topreal8.html#T23" target="_self">Th23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal8.html#E2:27"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topreal8.html#E3:27_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topreal8.html#E9:27_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="topreal8.html#E19:27_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="topreal8.html#E20:27_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topreal8.html#T23" target="_self">Th23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:27"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun2.html#T16">PARTFUN2:16</a></i>;<br/>


</div>
