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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>let </b><font color="Maroon">c<sub>3</sub></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="E1:64"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:64"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 ;<br/>

<a NAME="E4:64"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E3:64"><i><font color="Green">E39</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E5:64"/><b>then </b><i><font color="Green">E40</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>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:64"/><b>then </b><i><font color="Green">E41</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>assume </b><a NAME="E7:64"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E8:64"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="jordan3.html#T41">JORDAN3:41</a></i>;<br/>
<a NAME="E9:64"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></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="64_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><b>suppose </b></a><a NAME="E1:64_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1
 ;<br/><div class="add"><a NAME="E2:64_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="jordan3.html#K4">R_Cut</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>2</sub></font>,1,<span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="matrix_2.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan3.html#D5">JORDAN3:def 5</a></i>;<br/><a NAME="E3:64_1_1"/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>2</sub></font>,1,<span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:64"><i><font color="Green">E37</font></i></a>, <a class="txt" href="sprect_3.html#E6:64"><i><font color="Green">E41</font></i></a>, <a class="txt" href="sprect_3.html#E9:64"><i><font color="Green">E42</font></i></a>, <a class="ref" href="sprect_2.html#T26">SPRECT_2:26</a></i>;<br/><b>hence </b><a NAME="E4:64_1_1"/>
 <a href="jordan3.html#K4">R_Cut</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E2:64"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sprect_3.html#E2:64_1_1"><i><font color="Green">E43</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="64_1_2"><b>suppose </b></a><a NAME="E1:64_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1
 ;<br/><div class="add"><b>then </b> <a href="jordan3.html#K4">R_Cut</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="jordan3.html#D5">JORDAN3:def 5</a></i>
<br/>.= 
<span class="p1"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span>
<b>by </b><i><a class="txt" href="sprect_3.html#E6:64"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>2</sub></font>,1,1
<b>by </b><i><a class="txt" href="sprect_3.html#E5:64"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan4.html#T27">JORDAN4:27</a></i>
;<br/><b>hence </b><a NAME="E3:64_1_2"/>
 <a href="jordan3.html#K4">R_Cut</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="sprect_2.html#R1">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:64"><i><font color="Green">E37</font></i></a>, <a class="txt" href="sprect_3.html#E6:64"><i><font color="Green">E41</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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