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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>let </b><font color="Maroon">c<sub>2</sub></font>, <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="E1:27"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E14</font></i>: 
<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>1</sub></font>
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E15</font></i>: 
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> or (  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan3.html#R2">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:27"/><i><font color="Green">E17</font></i>: 
 <a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K4">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E1:27"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan5b.html#E2:27"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan5b.html#E3:27"><i><font color="Green">E15</font></i></a>, <a class="ref" href="jordan3.html#D8">JORDAN3:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1"/>
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> or (  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan3.html#R2">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan5b.html#E3:27"><i><font color="Green">E15</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>case </b></a><a NAME="E1:27_1_1_1"/>
 <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E1:27"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan5b.html#E2:27"><i><font color="Green">E14</font></i></a>, <a class="ref" href="jordan3.html#T64">JORDAN3:64</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2"><b>case </b></a><a NAME="E1:27_1_1_2"/>
(  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan3.html#R2">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2">Index</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:27_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan5b.html#E1:27"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan5b.html#E2:27"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan5b.html#E4:27"><i><font color="Green">E16</font></i></a>, <a class="ref" href="jordan3.html#T66">JORDAN3:66</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:27"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T17">SPPOL_2:17</a></i>;<br/>
<a NAME="E8:27"/><i><font color="Green">E19</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E5:27"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan5b.html#E7:27"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan3.html#T76">JORDAN3:76</a></i>;<br/>
<a NAME="E9:27"/>
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K3">L_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5b.html#E1:27"><i><font color="Green">E13</font></i></a>, <a class="ref" href="jordan3.html#T77">JORDAN3:77</a></i>;<br/>
<b>hence </b><a NAME="E10:27"/>
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K5">B_Cut</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan5b.html#E8:27"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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