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<b>let </b><font color="Maroon">f</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>

<b>consider </b><font color="Maroon">n1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">n2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:30"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>;<br/>
<a NAME="E4:30"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</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">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">df</font> =  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E52</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">df</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">df</font> <a href="hidden.html#R1">=</a> 0 or  <a href="card_1.html#K4">card</a> <font color="Maroon">df</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i>, <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="30_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">df</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">df</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:30_1_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E4:30_1_1_2"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:30_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:30"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>


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