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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="graph_2.html#V1" target="_self">TwoValued</a>
 ;<br/><a NAME="E2:27_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="graph_2.html#D3" target="_self">Def3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="card_2.html#T79">CARD_2:79</a></i>;<br/><b>thus </b><a NAME="E5:27_1"/>
 <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> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:27_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="graph_2.html#E1:27_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_2.html#E4:27_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_2"/><b>then </b>
1 <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>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_1_1_1_2"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><a NAME="E5:27_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E4:27_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:27_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="graph_2.html#E1:27_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E7:27_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E8:27_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E4:27_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T8">ZFMISC_1:8</a></i>;<br/><b>hence </b><a NAME="E9:27_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_2.html#E4:27_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E6:27_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="txt" href="graph_2.html#E4:27_1"><i><font color="Green">E22</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:27"/>
 <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> 1
 ;<br/>

<b>given </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E3:27"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> )
 ;<br/>

<a NAME="E4:27"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="txt" href="graph_2.html#E3:27"><i><font color="Green">E22</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>
<b>hence </b><a NAME="E5:27"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="graph_2.html#V1" target="_self">TwoValued</a>
 <b>by </b><i><a class="ref" href="graph_2.html#D3" target="_self">Def3</a></i>;<br/>


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