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<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">P</font> = <font color="Maroon" title="c2">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:35"/><i><font color="Green" title="E33">A2</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E1:35"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="jordan1c.html#T29" target="_self" title="JORDAN1C:th.29">Th29</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">l</font> = <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4;<br/>
<a NAME="E5:35"/>
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:35"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:35"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">n</font>
 <b>and </b><br/><a NAME="E8:35"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4
 <b>and </b><br/><a NAME="E9:35"/><i><font color="Green" title="E35">A4</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4
 <b>by </b><i><a class="ref" href="gobrd14.html#T21" title="GOBRD14:th.21">GOBRD14:21</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">n</font>
;<br/>

<b>set </b><font color="Maroon" title="c7">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font>;<br/>
<b>set </b><font color="Maroon" title="c8">W</font> =  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c9">E</font> =  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c10">S</font> =  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c11">N</font> =  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c12">EW</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">NS</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">I</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c15">J</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/>
<a NAME="E10:35"/><i><font color="Green" title="E36">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> &amp; <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1c.html#E1:35"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="jordan1c.html#T22" target="_self" title="JORDAN1C:th.22">Th22</a>, <a class="ref" href="jordan1c.html#T23" target="_self" title="JORDAN1C:th.23">Th23</a></i>;<br/>
<a NAME="E11:35"/><i><font color="Green" title="E37">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> &amp; <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1c.html#E1:35"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="jordan1c.html#T24" target="_self" title="JORDAN1C:th.24">Th24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">I</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>, <font color="Maroon" title="c17">J</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/>
<a NAME="E13:35"/><i><font color="Green" title="E38">A7</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan1c.html#T28" target="_self" title="JORDAN1C:th.28">Th28</a></i>;<br/>
<a NAME="E14:35"/><i><font color="Green" title="E39">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E15:35"/><i><font color="Green" title="E40">A9</font></i>: 
( <font color="Maroon" title="c16">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E16:35"/><b>then </b><i><font color="Green" title="E41">A10</font></i>: 
( <font color="Maroon" title="c16">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> &amp; <font color="Maroon" title="c17">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:35"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/>
<a NAME="E18:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E19:35"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E20:35"/><b>then </b><i><font color="Green" title="E42">A11</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="ref" href="jordan1c.html#T25" target="_self" title="JORDAN1C:th.25">Th25</a>, <a class="ref" href="jordan1c.html#T27" target="_self" title="JORDAN1C:th.27">Th27</a></i>;<br/>
<a NAME="E21:35"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E14:35"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="jordan1c.html#E16:35"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/>
<a NAME="E22:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E23:35"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E24:35"/><b>then </b><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="ref" href="jordan1c.html#T26" target="_self" title="JORDAN1C:th.26">Th26</a></i>;<br/>
<a NAME="E25:35"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">I</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E14:35"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="jordan1c.html#E16:35"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/>
<a NAME="E26:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E27:35"/><b>then </b><i><font color="Green" title="E44">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E28:35"/><b>then </b><i><font color="Green" title="E45">A14</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan1c.html#T28" target="_self" title="JORDAN1C:th.28">Th28</a></i>;<br/>
<a NAME="E29:35"/><i><font color="Green" title="E46">A15</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E13:35"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="jordan1c.html#E20:35"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="jordan1c.html#E24:35"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="jordan1c.html#E27:35"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<a NAME="E31:35"/>
<font color="Maroon" title="c18">GIJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/>
<a NAME="E32:35"/><b>then </b><i><font color="Green" title="E47">A16</font></i>: 
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E8:35"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="jordan1c.html#E9:35"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E29:35"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="jordan1c.html#T30" target="_self" title="JORDAN1C:th.30">Th30</a></i>;<br/>
<a NAME="E33:35"/><i><font color="Green" title="E48">A17</font></i>: 
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:35"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="jordan1c.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/>
<a NAME="E34:35"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E32:35"><i><font color="Green" title="E47">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:35"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">GIJ</font>,<font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E36:35"/><b>then </b><i><font color="Green" title="E49">A18</font></i>: 
<font color="Maroon" title="c18">GIJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<a NAME="E38:35"/>
<font color="Maroon" title="c19">GIJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/>
<a NAME="E39:35"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E8:35"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="jordan1c.html#E9:35"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E29:35"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="jordan1c.html#T30" target="_self" title="JORDAN1C:th.30">Th30</a></i>;<br/>
<a NAME="E40:35"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E33:35"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E41:35"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c19">GIJ</font>,<font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E42:35"/><b>then </b><i><font color="Green" title="E50">A19</font></i>: 
<font color="Maroon" title="c19">GIJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E9"><i><font color="Green" title="E7">Lm5</font></i></a></i>;<br/>
<a NAME="E44:35"/>
<font color="Maroon" title="c20">GIJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/>
<a NAME="E45:35"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E8:35"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="jordan1c.html#E9:35"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E29:35"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="jordan1c.html#T30" target="_self" title="JORDAN1C:th.30">Th30</a></i>;<br/>
<a NAME="E46:35"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E33:35"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E47:35"/><b>then </b><i><font color="Green" title="E51">A20</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">GIJ</font>,<font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E9"><i><font color="Green" title="E7">Lm5</font></i></a></i>;<br/>
<a NAME="E49:35"/>
<font color="Maroon" title="c21">GIJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/>
<a NAME="E50:35"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E8:35"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="jordan1c.html#E9:35"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E29:35"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="jordan1c.html#T30" target="_self" title="JORDAN1C:th.30">Th30</a></i>;<br/>
<a NAME="E51:35"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E33:35"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E52:35"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c21">GIJ</font>,<font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/>
<a NAME="E53:35"/><b>then </b><i><font color="Green" title="E52">A21</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font> )
 
<b>by </b><i><a class="txt" href="jordan1c.html#E47:35"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/><i><font color="Green" title="E53">A22</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c16">I</font>
;<br/><b>take </b>
<font color="Maroon" title="c17">J</font>
;<br/><b>thus </b><a NAME="E2:35_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">I</font> &amp; <font color="Maroon" title="c16">I</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">J</font> &amp; <font color="Maroon" title="c17">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E15:35"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:35_1_1"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E4:35"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="jordan1c.html#E8:35"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="jordan1c.html#E9:35"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E29:35"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="jordan1c.html#E36:35"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="jordan1c.html#E42:35"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="jordan1c.html#E53:35"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="jordan1c.html#E10"><i><font color="Green" title="E8">Lm6</font></i></a></i>;<br/><b>hence </b><a NAME="E4:35_1_1"/>
( <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E4:35"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E13:35"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="jordan1c.html#E20:35"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="jordan1c.html#E24:35"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="jordan1c.html#E27:35"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="jordan1c.html#E1:35_1_1"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/><i><font color="Green" title="E53">A23</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c22">q</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font>;<br/><b>reconsider </b><font color="Maroon" title="c23">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/><a NAME="E3:35_1_2"/><i><font color="Green" title="E54">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">I</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E4:35_1_2"/><i><font color="Green" title="E55">A25</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">J</font> &amp; <font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E1:35"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E16:35"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="jordan1c.html#T24" target="_self" title="JORDAN1C:th.24">Th24</a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E5:35_1_2"/><b>then </b><i><font color="Green" title="E56">A26</font></i>: 
<font color="Maroon" title="c23">GIJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/><a NAME="E6:35_1_2"/><i><font color="Green" title="E57">A27</font></i>: 
<font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">I</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E4:35_1_2"><i><font color="Green" title="E55">A25</font></i></a>, <a class="ref" href="binarith.html#T69" title="BINARITH:th.69">BINARITH:69</a></i>;<br/><a NAME="E7:35_1_2"/><b>then </b><i><font color="Green" title="E58">A28</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E16:35"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="jordan1c.html#E1:35_1_2"><i><font color="Green" title="E53">A23</font></i></a>, <a class="txt" href="jordan1c.html#E4:35_1_2"><i><font color="Green" title="E55">A25</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/><a NAME="E8:35_1_2"/><i><font color="Green" title="E59">A29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan1c.html#E3:35_1_2"><i><font color="Green" title="E54">A24</font></i></a>, <a class="txt" href="jordan1c.html#E4:35_1_2"><i><font color="Green" title="E55">A25</font></i></a>, <a class="ref" href="jordan9.html#T18" title="JORDAN9:th.18">JORDAN9:18</a></i>;<br/><a NAME="E9:35_1_2"/><i><font color="Green" title="E60">A30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan1c.html#E3:35_1_2"><i><font color="Green" title="E54">A24</font></i></a>, <a class="txt" href="jordan1c.html#E4:35_1_2"><i><font color="Green" title="E55">A25</font></i></a>, <a class="ref" href="jordan9.html#T18" title="JORDAN9:th.18">JORDAN9:18</a></i>;<br/><a NAME="E10:35_1_2"/><b>then </b><i><font color="Green" title="E61">A31</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E20:35"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="jordan1c.html#E28:35"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="jordan1c.html#E4:35_1_2"><i><font color="Green" title="E55">A25</font></i></a>, <a class="txt" href="jordan1c.html#E7:35_1_2"><i><font color="Green" title="E58">A28</font></i></a>, <a class="txt" href="jordan1c.html#E8:35_1_2"><i><font color="Green" title="E59">A29</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></i>;<br/><a NAME="E11:35_1_2"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="jordan1c.html#E8:35"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="jordan1c.html#E9:35"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1c.html#E4:35_1_2"><i><font color="Green" title="E55">A25</font></i></a>, <a class="txt" href="jordan1c.html#E5:35_1_2"><i><font color="Green" title="E56">A26</font></i></a>, <a class="ref" href="jordan1c.html#T30" target="_self" title="JORDAN1C:th.30">Th30</a></i>;<br/><a NAME="E12:35_1_2"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E33:35"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:35_1_2"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">GIJ</font>,<font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/><a NAME="E14:35_1_2"/><b>then </b><i><font color="Green" title="E62">A32</font></i>: 
<font color="Maroon" title="c23">GIJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><b>take </b>
<font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
;<br/><b>take </b>
<font color="Maroon" title="c17">J</font>
;<br/><a NAME="E15:35_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/><a NAME="E16:35_1_2"/><b>then </b><i><font color="Green" title="E63">A33</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">J</font> &amp; <font color="Maroon" title="c17">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E11:35"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="jordan1c.html#E15:35"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:35_1_2"/>
<font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_2_1"/>
<font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E2:35_1_2_1"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">I</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/>
<a NAME="E3:35_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E16:35_1_2"><i><font color="Green" title="E63">A33</font></i></a>, <a class="ref" href="jordan8.html#T14" title="JORDAN8:th.14">JORDAN8:14</a></i>;<br/>
<a NAME="E4:35_1_2_1"/><b>then </b><i><font color="Green" title="E64">A34</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E1:35"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="jordan1c.html#E1:35_1_2"><i><font color="Green" title="E53">A23</font></i></a>, <a class="ref" href="jordan1c.html#T18" target="_self" title="JORDAN1C:th.18">Th18</a></i>;<br/>
<a NAME="E5:35_1_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E1:35"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="jordan1c.html#T34" target="_self" title="JORDAN1C:th.34">Th34</a></i>;<br/>
<a NAME="E6:35_1_2_1"/><b>then </b><i><font color="Green" title="E65">A35</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:35_1_2_1"><i><font color="Green" title="E64">A34</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:35_1_2_1"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T114" title="JORDAN2C:th.114">JORDAN2C:114</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1c.html#E1:35"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="jordan1c.html#E6:35_1_2_1"><i><font color="Green" title="E65">A35</font></i></a>, <a class="ref" href="jordan1c.html#T6" target="_self" title="JORDAN1C:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:35_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">J</font> &amp; <font color="Maroon" title="c17">J</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E1:35"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="jordan1c.html#E15:35"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="jordan1c.html#E16:35"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="jordan1c.html#E4:35_1_2"><i><font color="Green" title="E55">A25</font></i></a>, <a class="txt" href="jordan1c.html#E6:35_1_2"><i><font color="Green" title="E57">A27</font></i></a>, <a class="ref" href="jordan1c.html#T24" target="_self" title="JORDAN1C:th.24">Th24</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c24">q</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c25">GIJ</font> = <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/><a NAME="E20:35_1_2"/><i><font color="Green" title="E64">A36</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">J</font> &amp; <font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E1:35"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E16:35"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="jordan1c.html#T24" target="_self" title="JORDAN1C:th.24">Th24</a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E21:35_1_2"/><b>then </b><i><font color="Green" title="E65">A37</font></i>: 
<font color="Maroon" title="c25">GIJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/><a NAME="E22:35_1_2"/>
<font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">I</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E20:35_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="binarith.html#T69" title="BINARITH:th.69">BINARITH:69</a></i>;<br/><a NAME="E23:35_1_2"/><b>then </b><i><font color="Green" title="E66">A38</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E16:35"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="jordan1c.html#E1:35_1_2"><i><font color="Green" title="E53">A23</font></i></a>, <a class="txt" href="jordan1c.html#E20:35_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/><a NAME="E24:35_1_2"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E20:35"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="jordan1c.html#E28:35"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="jordan1c.html#E8:35_1_2"><i><font color="Green" title="E59">A29</font></i></a>, <a class="txt" href="jordan1c.html#E9:35_1_2"><i><font color="Green" title="E60">A30</font></i></a>, <a class="txt" href="jordan1c.html#E20:35_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></i>;<br/><a NAME="E25:35_1_2"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="jordan1c.html#E8:35"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="jordan1c.html#E9:35"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1c.html#E20:35_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="jordan1c.html#E21:35_1_2"><i><font color="Green" title="E65">A37</font></i></a>, <a class="ref" href="jordan1c.html#T30" target="_self" title="JORDAN1C:th.30">Th30</a></i>;<br/><a NAME="E26:35_1_2"/><b>then </b>
 <a href="topreal6.html#K1" title="TOPREAL6:func.1">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E33:35"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E27:35_1_2"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c25">GIJ</font>,<font color="Maroon" title="c3">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="topreal6.html#D1" title="TOPREAL6:def.1">TOPREAL6:def 1</a></i>;<br/><a NAME="E28:35_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E10:35"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1c.html#E36:35"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="jordan1c.html#E53:35"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><a NAME="E29:35_1_2"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="jordan1c.html#E4:35"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="jordan1c.html#E8:35"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="jordan1c.html#E9:35"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1c.html#E10:35_1_2"><i><font color="Green" title="E61">A31</font></i></a>, <a class="txt" href="jordan1c.html#E14:35_1_2"><i><font color="Green" title="E62">A32</font></i></a>, <a class="txt" href="jordan1c.html#E20:35_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="jordan1c.html#E10"><i><font color="Green" title="E8">Lm6</font></i></a></i>;<br/><b>hence </b><a NAME="E30:35_1_2"/>
( <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c6">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan1c.html#E4:35"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="jordan1c.html#E20:35"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="jordan1c.html#E28:35"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="jordan1c.html#E8:35_1_2"><i><font color="Green" title="E59">A29</font></i></a>, <a class="txt" href="jordan1c.html#E9:35_1_2"><i><font color="Green" title="E60">A30</font></i></a>, <a class="txt" href="jordan1c.html#E20:35_1_2"><i><font color="Green" title="E64">A36</font></i></a>, <a class="txt" href="jordan1c.html#E23:35_1_2"><i><font color="Green" title="E66">A38</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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