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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="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="topreal2.html#V1">being_simple_closed_curve</a> <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</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>4</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="E1:28"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E3:28"/><i><font color="Green">E23</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:28"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E6:28"/><i><font color="Green">E26</font></i>: 
 <a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan1f.html#T12">JORDAN1F:12</a></i>;<br/>
<a NAME="E7:28"/><i><font color="Green">E27</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan1b.html#T12">JORDAN1B:12</a>, <a class="ref" href="jordan1b.html#T14">JORDAN1B:14</a></i>;<br/>
<a NAME="E8:28"/><i><font color="Green">E28</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> &amp;  <a href="jordan1a.html#K1">Center</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan1b.html#T12">JORDAN1B:12</a>, <a class="ref" href="jordan1b.html#T14">JORDAN1B:14</a></i>;<br/>
<a NAME="E9:28"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 
<b>by </b><i><a class="ref" href="jordan8.html#T13">JORDAN8:13</a></i>;<br/>
<a NAME="E10:28"/><b>then </b>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E11:28"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:28"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E7:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E13:28"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan1i.html#E7:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E14:28"/>
 <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan6.html#T76">JORDAN6:76</a></i>;<br/>
<a NAME="E15:28"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan1i.html#E4:28"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan1i.html#E7:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="jordan1g.html#T54">JORDAN1G:54</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:28"/><i><font color="Green">E31</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan1i.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan1i.html#E6:28"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan1i.html#E12:28"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan1i.html#E13:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan1f.html#T2">JORDAN1F:2</a></i>;<br/>
<a NAME="E18:28"/>
4 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan8.html#T13">JORDAN8:13</a></i>;<br/>
<a NAME="E19:28"/><b>then </b><i><font color="Green">E32</font></i>: 
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="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:28"/>
4 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan8.html#T13">JORDAN8:13</a></i>;<br/>
<a NAME="E21:28"/><b>then </b>
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="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:28"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1i.html#E2:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1i.html#E19:28"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan1a.html#T57">JORDAN1A:57</a></i>;<br/>
<a NAME="E23:28"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1i.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan1i.html#E7:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/>
<a NAME="E24:28"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E2:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E25:28"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1i.html#E7:28"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan1i.html#E8:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1a.html#T64">JORDAN1A:64</a></i>;<br/>
<a NAME="E26:28"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1i.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan1i.html#E7:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="jordan1a.html#T40">JORDAN1A:40</a></i>;<br/>
<a NAME="E27:28"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E22:28"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1i.html#E23:28"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1i.html#E25:28"><i><font color="Green">E35</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<a NAME="E28:28"/>
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E29:28"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E27:28"><i><font color="Green">E36</font></i></a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<a NAME="E30:28"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E31:28"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:28_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <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> <b>by </b><i><a class="txt" href="jordan1i.html#E1:28_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E4:28_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1i.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan1i.html#E7:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="jordan1a.html#T40">JORDAN1A:40</a></i>;<br/>
<a NAME="E5:28_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1i.html#E25:28"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:28_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jordan1i.html#E2:28_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E7:28_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1i.html#E22:28"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1i.html#E23:28"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1i.html#E2:28_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/>
<a NAME="E8:28_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K1">Upper_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E2:28_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:28_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan1e.html#T17">JORDAN1E:17</a></i>;<br/>
<a NAME="E10:28_1"/><b>then </b>
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E11:28_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1i.html#E7:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="jordan1a.html#T93">JORDAN1A:93</a></i>;<br/>
<a NAME="E12:28_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E23:28"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1i.html#E6:28_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan1i.html#E7:28_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<b>hence </b><a NAME="E13:28_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1i.html#E2:28_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:28"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E30:28"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E33:28"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<span class="default"><a href="jordan1e.html#K2">Lower_Seq</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span></span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p5">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1i.html#E2:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="jordan1g.html#T64">JORDAN1G:64</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E34:28"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan1i.html#E17:28"><i><font color="Green">E31</font></i></a></i>;<br/>

<b>thus </b><a NAME="E35:28"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1i.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan1i.html#E17:28"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<a NAME="E36:28"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1i.html#E1:28"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan1i.html#E2:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1i.html#E17:28"><i><font color="Green">E31</font></i></a>, <a class="ref" href="jordan1g.html#T43">JORDAN1G:43</a></i>;<br/>
<b>hence </b><a NAME="E37:28"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1a.html#K1">Center</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan1i.html#E5:28"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan1i.html#E17:28"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1i.html#E33:28"><i><font color="Green">E38</font></i></a>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


</div>
