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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="finseq_1.html#NM4">FinSubsequence</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:3"/><i><font color="Green">E4</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/>
<b>thus </b><a NAME="E3:3"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:3_1"/><i><font color="Green">E6</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/>
<a NAME="E3:3_1"/>
 <a href="relat_1.html#K1">dom</a> <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="txt" href="pnproc_1.html#E1:3_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<b>hence </b> <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>1</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="pnproc_1.html#E2:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="pnproc_1.html#E2:3_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_1.html#T72">FINSEQ_1:72</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:3"/>
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <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="txt" href="pnproc_1.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>


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