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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/>



<b>thus </b><a NAME="E1:5"/>
( <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_2.html#K2">|-&gt;</a> 0 implies  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>

<b>assume </b><a NAME="E2:5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:5"/><b>then </b><i><font color="Green">E3</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1"/>
(  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1"><b>suppose </b></a><a NAME="E1:5_1_1"/><i><font color="Green">E4</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>2</sub></font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="mssublat.html#E3:5"><i><font color="Green">E3</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>
<br/>.= 
0 <a href="finseq_2.html#K2">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> 0
<b>by </b><i><a class="txt" href="mssublat.html#E1:5_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2"><b>suppose </b></a><a NAME="E1:5_1_2"/><i><font color="Green">E4</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E3:5_1_2"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> or  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><a NAME="E4:5_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0
 <b>by </b><i><a class="txt" href="mssublat.html#E3:5"><i><font color="Green">E3</font></i></a>, <a class="txt" href="mssublat.html#E1:5_1_2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="funcop_1.html#T15">FUNCOP_1:15</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><b>hence </b><a NAME="E5:5_1_2"/>
<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_2.html#K2">|-&gt;</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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