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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:31"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green">E15</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E3:31"/><b>then </b><i><font color="Green">E16</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green">E17</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E18</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E2:31"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green">E19</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E2:31"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E1:31"><i><font color="Green">E14</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="jordan5b.html#E4:31"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E9:31"/><i><font color="Green">E22</font></i>: 
 <a href="jordan3.html#K2">Index</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan5b.html#E2:31"><i><font color="Green">E15</font></i></a>, <a class="ref" href="jordan3.html#T44">JORDAN3:44</a></i>;<br/>
<a NAME="E10:31"/>
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E4:31"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan5b.html#E5:31"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5b.html#E7:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/>
<a NAME="E11:31"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E5:31"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b> <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan5b.html#E9:31"><i><font color="Green">E22</font></i></a>, <a class="ref" href="jordan3.html#D4">JORDAN3:def 4</a></i>
<br/>.= 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan5b.html#E4:31"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan5b.html#E6:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan5b.html#E8:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan5b.html#E9:31"><i><font color="Green">E22</font></i></a>, <a class="ref" href="jordan3.html#T56">JORDAN3:56</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="jordan5b.html#E3:31"><i><font color="Green">E16</font></i></a>, <a class="ref" href="jordan3.html#T29">JORDAN3:29</a></i>
;<br/><br/>


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