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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:37"/><i><font color="Green">E20</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>case </b></a><a NAME="E1:37_1_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"> <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>2</sub></font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>2</sub></font>,1,<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="finseq_8.html#E1:37"><i><font color="Green">E20</font></i></a>, <a class="txt" href="finseq_8.html#E1:37_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_8.html#D8" target="_self">Def8</a></i>
;<br/><b>hence </b><a NAME="E3:37_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">c<sub>2</sub></font>,1
 <b>by </b><i><a class="txt" href="finseq_8.html#E1:37"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_8.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><b>case </b></a><a NAME="E1:37_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">c<sub>2</sub></font>,1
 <b>by </b><i><a class="ref" href="finseq_8.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">c<sub>2</sub></font>,1
 ;<br/>


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