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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="finseq_1.html#NM4">FinSubsequence</a>;<br/>



<b>assume </b><a NAME="E1:119"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="finseq_1.html#NM4">FinSubsequence</a><b> such that </b><br/><a NAME="E3:119"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:119"/><i><font color="Green">E83</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:119"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#T81" target="_self">Th81</a></i>;<br/>
<a NAME="E6:119"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">b<sub>1</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"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E5:119"><i><font color="Green">E84</font></i></a>, <a class="ref" href="grfunc_1.html#T34">GRFUNC_1:34</a></i>;<br/>
<a NAME="E7:119"/>
for <font color="Olive">b<sub>1</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"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:119_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:119_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <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"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pnproc_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E3:119_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E1:119_1"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pnproc_1.html#E5:119"><i><font color="Green">E84</font></i></a>, <a class="ref" href="grfunc_1.html#T34">GRFUNC_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="pnproc_1.html#E1:119_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="pnproc_1.html#D15" target="_self">Def15</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E8:119"/><b>then </b>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E4:119"><i><font color="Green">E83</font></i></a>, <a class="txt" href="pnproc_1.html#E6:119"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:119"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E3:119"><i><font color="Green">E82</font></i></a></i>;<br/>


</div>
