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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="connsp_1.html#V2">connected</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/>



<a NAME="E1:25"/><i><font color="Green">E11</font></i>: 
 <a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan9.html#D1">JORDAN9:def 1</a></i>;<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><b>that </b><br/><a NAME="E2:25"/><i><font color="Green">E12</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green">E13</font></i>: 
<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="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:25"/><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:25"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan9.html#T34">JORDAN9:34</a></i>;<br/>
<a NAME="E6:25"/><b>then </b><i><font color="Green">E16</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</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="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</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="sprect_2.html#T73">SPRECT_2:73</a></i>;<br/>
<a NAME="E7:25"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</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="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</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="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</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="jordan1e.html#E5:25"><i><font color="Green">E15</font></i></a>, <a class="ref" href="sprect_2.html#T74">SPRECT_2:74</a></i>;<br/>
<a NAME="E8:25"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</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="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan1e.html#E6:25"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:25"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</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="jordan1e.html#E3:25"><i><font color="Green">E13</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:25"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</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="jordan1e.html#E2:25"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:25"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</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="finseq_4.html#K5">..</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan1e.html#E4:25"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_5.html#T42">FINSEQ_5:42</a></i>;<br/>
<a NAME="E12:25"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan1e.html#E8:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:25"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E15:25"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</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>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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E16:25"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E17:25"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</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="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E18:25"/><i><font color="Green">E25</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan1e.html#E1:25"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan1e.html#E3:25"><i><font color="Green">E13</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E19:25"/><i><font color="Green">E26</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</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> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1e.html#E14:25"><i><font color="Green">E21</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E20:25"/><i><font color="Green">E27</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</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> &amp; 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>  <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="txt" href="jordan1e.html#E16:25"><i><font color="Green">E23</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E21:25"/><i><font color="Green">E28</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/>
<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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
 <a href="pscomp_1.html#K17">E-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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan1e.html#E4:25"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan1e.html#E15:25"><i><font color="Green">E22</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</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> <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>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1e.html#E19:25"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan1e.html#E21:25"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1a.html#T92">JORDAN1A:92</a></i>
;<br/>
<a NAME="E23:25"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</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> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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="jordan1e.html#E19:25"><i><font color="Green">E26</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/>
<a NAME="E24:25"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="jordan1e.html#E10:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_3.html#T28">FINSEQ_3:28</a></i>;<br/>
<a NAME="E26:25"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan1e.html#E12:25"><i><font color="Green">E20</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E27:25"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E28:25"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</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="jordan1e.html#E3:25"><i><font color="Green">E13</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E30:25"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E31:25"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</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>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>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E32:25"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E33:25"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</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="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E34:25"/><i><font color="Green">E37</font></i>: 
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan1e.html#E1:25"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan1e.html#E9:25"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan1e.html#E27:25"><i><font color="Green">E31</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E35:25"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E36:25"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan1e.html#E14:25"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan1e.html#E15:25"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1e.html#E32:25"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan1e.html#E33:25"><i><font color="Green">E36</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<a NAME="E37:25"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</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> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1e.html#E30:25"><i><font color="Green">E33</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E38:25"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan1e.html#E34:25"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan1e.html#E1:25_2"><i><font color="Green">E42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1"><b>suppose </b></a><a NAME="E1:25_2_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1e.html#E9:25"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan1e.html#E23:25"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan1e.html#E27:25"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1e.html#E30:25"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan1e.html#E31:25"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1e.html#E32:25"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan1e.html#E33:25"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan1e.html#E36:25"><i><font color="Green">E39</font></i></a>, <a class="ref" href="jordan10.html#T1">JORDAN10:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2"><b>suppose </b></a><a NAME="E1:25_2_1_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:25_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</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="jordan1e.html#E28:25"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:25_2_1_2"/><b>then </b>
<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>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K25">E-most</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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1e.html#E24:25"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan1e.html#E27:25"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1e.html#E31:25"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1e.html#E36:25"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan1e.html#E37:25"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan1e.html#E1:25_2_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan1a.html#T82">JORDAN1A:82</a></i>;<br/><a NAME="E4:25_2_1_2"/><b>then </b><i><font color="Green">E44</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>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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>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>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan1e.html#E4:25"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan1e.html#E33:25"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan1e.html#E1:25_2_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="pscomp_1.html#T108">PSCOMP_1:108</a></i>;<br/><a NAME="E5:25_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E6:25_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1e.html#E19:25"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan1e.html#E36:25"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan1e.html#E37:25"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan1e.html#E1:25_2_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan1e.html#E4:25_2_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E39:25"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_3_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan1e.html#E18:25"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan1e.html#E1:25_3"><i><font color="Green">E42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_1"><b>suppose </b></a><a NAME="E1:25_3_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1e.html#E20:25"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan1e.html#E23:25"><i><font color="Green">E29</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_2"><b>suppose </b></a><a NAME="E1:25_3_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><div class="add"><a NAME="E2:25_3_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan1e.html#E17:25"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan1e.html#E23:25"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan1e.html#E31:25"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan1e.html#E34:25"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan1e.html#E36:25"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan1e.html#E37:25"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan1e.html#E38:25"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:25_3_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 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="jordan9.html#K1">Cage</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="jordan1e.html#E3:25"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E4:25_3_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan1e.html#E27:25"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1e.html#E35:25"><i><font color="Green">E38</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E5:25_3_1_2"/>
( <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan1e.html#E2:25"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan1e.html#E3:25"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan1e.html#E9:25"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan1e.html#E27:25"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan1e.html#E35:25"><i><font color="Green">E38</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/><a NAME="E6:25_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan1e.html#E2:25_3_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan1e.html#E4:25_3_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:25_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E8:25_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1e.html#E14:25"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan1e.html#E15:25"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1e.html#E16:25"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan1e.html#E17:25"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan1e.html#E1:25_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></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>6</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1e.html#E19:25"><i><font color="Green">E26</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>6</sub></font>,<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="jordan1e.html#E20:25"><i><font color="Green">E27</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/>
<b>hence </b><a NAME="E41:25"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1e.html#E4:25"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan1e.html#E15:25"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan1e.html#E17:25"><i><font color="Green">E24</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>


</div>
