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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>, <font color="Maroon">g</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><b>that </b><br/><a NAME="E1:62"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">g</font> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:62"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">g</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 ;<br/>

<a NAME="E4:62"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E3:62"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E5:62"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:62"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>assume </b><a NAME="E7:62"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E8:62"/><b>then </b>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E9:62"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font></span>)</span> <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">g</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:62"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E11:62"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font></span>)</span> <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">g</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><b>suppose </b></a><a NAME="E1:62_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:62_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="matrix_2.html#K4">*&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">g</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</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">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#D4">JORDAN3:def 4</a></i>;<br/><a NAME="E3:62_1_1"/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:62"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E6:62"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="sprect_2.html#T26">SPRECT_2:26</a></i>;<br/><b>hence </b><a NAME="E4:62_1_1"/>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E2:62"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="sprect_2.html#T28">SPRECT_2:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_2"><b>suppose </b></a><a NAME="E1:62_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:62_1_2"/><b>then </b>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">p</font>,<font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#D4">JORDAN3:def 4</a></i>;<br/><b>hence </b><a NAME="E3:62_1_2"/>
 <a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">g</font>,<font color="Maroon">p</font> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:62"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E6:62"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="sprect_2.html#T26">SPRECT_2:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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