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<b>let </b><font color="Maroon" title="c1">f</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="jordan23.html#V2" title="JORDAN23:attr.2">weakly-one-to-one</a>
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green" title="E32">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ;<br/>

<a NAME="E3:59"/><i><font color="Green" title="E33">A3</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:59"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:59"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:59"/>
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E6:59"/><b>then </b><i><font color="Green" title="E35">A5</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:59"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:59"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E4:59"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E8:59"/><i><font color="Green" title="E37">A7</font></i>: 
 <a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan23.html#E2:59"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="jordan3.html#T44" title="JORDAN3:th.44">JORDAN3:44</a></i>;<br/>
<a NAME="E9:59"/>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E1:59"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jordan23.html#E6:59"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="jordan23.html#D2" target="_self" title="JORDAN23:def.2">Def2</a></i>;<br/>
<a NAME="E10:59"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan23.html#E4:59"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b> <a href="jordan3.html#K3" title="JORDAN3:func.3">L_Cut</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> = 
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan23.html#E8:59"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="jordan3.html#D4" title="JORDAN3:def.4">JORDAN3:def 4</a></i>
<br/>.= 
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan23.html#E4:59"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan23.html#E6:59"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="jordan23.html#E7:59"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jordan23.html#E8:59"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="jordan3.html#T56" title="JORDAN3:th.56">JORDAN3:56</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font>
<b>by </b><i><a class="txt" href="jordan23.html#E3:59"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="jordan3.html#T29" title="JORDAN3:th.29">JORDAN3:29</a></i>
;<br/><br/>


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