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<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</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="E1:48"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>

<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:48"/><i><font color="Green">E51</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <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">f</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="jordan3.html#E5:48"><i><font color="Green">E52</font></i></a>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/>
<a NAME="E4:48"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E6:48"><i><font color="Green">E53</font></i></a>, </i>;<br/>
<a NAME="E5:48"/><b>then </b><i><font color="Green">E52</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E6:48"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:48"/><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> &amp;  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="txt" href="jordan3.html#E5:48"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan3.html#T4" target="_self">Th4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><b>case </b></a><a NAME="E1:48_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1
 ;<br/><div class="add"><a NAME="E2:48_1_1_1"/><b>then </b>
 <a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,1,<span class="p2">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/><b>hence </b> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,1,<span class="p3">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">f</font>,1,<span class="p3">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="jordan3.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="jordan3.html#T17" target="_self">Th17</a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2"><b>case </b></a><a NAME="E1:48_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1
 ;<br/><div class="add"><a NAME="E2:48_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:48_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E6:48"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:48_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="jordan3.html#E6:48"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:48_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:48_1_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E7:48_1_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E8:48_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>thus </b> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> = 
1
<b>by </b><i><a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
 <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan3.html#T21" target="_self">Th21</a>, <a class="ref" href="jordan3.html#T9" target="_self">Th9</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:48"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 implies  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 ) &amp; ( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 implies  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K4" target="_self">R_Cut</a> <font color="Maroon">f</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2" target="_self">Index</a> <font color="Maroon">p</font>,<font color="Maroon">f</font> ) )
 ;<br/>


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