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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E16</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>

<i><font color="Green">E17</font></i>: <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2 = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="ref" href="jordan8.html#T16">JORDAN8:16</a></i>
;<br/>
<i><font color="Green">E18</font></i>: <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="ref" href="jordan8.html#T17">JORDAN8:17</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</span>)</span> <a href="euclid.html#K21">`1</a> ;<br/>
<a NAME="E4:27"/><i><font color="Green">E19</font></i>: 
 <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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green">E20</font></i>: 
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>2</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="E6:27"/><i><font color="Green">E21</font></i>: 
1 <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="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E5:27"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:27"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E8:27"/><b>then </b><i><font color="Green">E22</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E9:27"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/>
<i><font color="Green">E24</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</span>)</span> <a href="euclid.html#K21">`1</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan1b.html#E4:27"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan1b.html#E6:27"><i><font color="Green">E21</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan1b.html#E4:27"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan1b.html#E8:27"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1b.html#E9:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/>
<a NAME="E11:27"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:27"/><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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 2,2</span>)</span> <a href="euclid.html#K21">`1</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan1b.html#E4:27"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan1b.html#E6:27"><i><font color="Green">E21</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/>
<a NAME="E13:27"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>2</sub></font> <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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1b.html#E6:27"><i><font color="Green">E21</font></i></a>, <a class="ref" href="jordan8.html#T14">JORDAN8:14</a></i>;<br/>
<a NAME="E14:27"/>
<font color="Maroon">c<sub>3</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="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E15:27"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E16:27"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E17:27"/><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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</span>)</span> <a href="euclid.html#K21">`1</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>2</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan1b.html#E4:27"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan1b.html#E8:27"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1b.html#E9:27"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan1b.html#E10:27"><i><font color="Green">E24</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/>
<a NAME="E18:27"/><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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E8:27"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1b.html#E9:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="jordan8.html#T15">JORDAN8:15</a></i>;<br/>
<a NAME="E19:27"/><b>then </b><i><font color="Green">E26</font></i>: 
 <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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E2:27"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan1b.html#E3:27"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan1b.html#E10:27"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan1b.html#E13:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="jordan1b.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E20:27"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2 <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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan1b.html#E4:27"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan1b.html#E6:27"><i><font color="Green">E21</font></i></a>, <a class="ref" href="jordan1a.html#T37">JORDAN1A:37</a></i>;<br/>
<a NAME="E21:27"/>
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:27"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan1b.html#E4:27"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan1b.html#E8:27"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1b.html#E9:27"><i><font color="Green">E23</font></i></a>, <a class="ref" href="jordan1a.html#T37">JORDAN1A:37</a></i>;<br/>
<a NAME="E22:27"/><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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,2</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E20:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="topreal1.html#T12">TOPREAL1:12</a></i>;<br/>
<b>hence </b><a NAME="E23:27"/>
 <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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan1b.html#E19:27"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


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