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<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> non <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> non <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<a NAME="E1:32"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">W-most</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T91" title="PSCOMP_1:th.91">PSCOMP_1:91</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">p</font> being   <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><b> such that </b><br/><a NAME="E3:32"/><i><font color="Green" title="E27">A1</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="jordan1a.html#T110" title="JORDAN1A:th.110">JORDAN1A:110</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1d.html#E3:32"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:32"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K9" title="TOPREAL1:func.9">west_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font></span>)</span> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">i</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="E7:32"/><i><font color="Green" title="E30">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1: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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E8:32"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">i</font>
;<br/>

<b>thus </b><a NAME="E9:32"/><i><font color="Green" title="E32">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan1d.html#E7:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>

<a NAME="E10:32"/><i><font color="Green" title="E33">A7</font></i>: 
 <a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="jordan9.html#D1" title="JORDAN9:def.1">JORDAN9:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">i1</font>, <font color="Maroon" title="c6">j1</font>, <font color="Maroon" title="c7">i2</font>, <font color="Maroon" title="c8">j2</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="E12:32"/><i><font color="Green" title="E34">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">j1</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E13:32"/><i><font color="Green" title="E35">A9</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">j1</font>
 <b>and </b><br/><a NAME="E14:32"/><i><font color="Green" title="E36">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c8">j2</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E15:32"/><i><font color="Green" title="E37">A11</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i2</font>,<font color="Maroon" title="c8">j2</font>
 <b>and </b><br/><a NAME="E16:32"/><i><font color="Green" title="E38">A12</font></i>: 
( ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> ) or ( <font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan1d.html#E7:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<a NAME="E17:32"/><i><font color="Green" title="E39">A13</font></i>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">W-most</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T91" title="PSCOMP_1:th.91">PSCOMP_1:91</a></i>;<br/>
<a NAME="E18:32"/><b>then </b><i><font color="Green" title="E40">A14</font></i>: 
<font color="Maroon" title="c3">p</font> <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#K13" title="PSCOMP_1:func.13">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1d.html#E4:32"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="jordan1a.html#T106" title="JORDAN1A:th.106">JORDAN1A:106</a></i>;<br/>
<a NAME="E19:32"/><i><font color="Green" title="E41">A15</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1d.html#E7:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E20:32"/><b>then </b><i><font color="Green" title="E42">A16</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1d.html#E5:32"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan1d.html#E8:32"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="jordan1d.html#E9:32"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan1d.html#E17:32"><i><font color="Green" title="E39">A13</font></i></a>, <a class="ref" href="jordan1a.html#T102" title="JORDAN1A:th.102">JORDAN1A:102</a>, <a class="ref" href="sppol_1.html#T64" title="SPPOL_1:th.64">SPPOL_1:64</a></i>;<br/>
<a NAME="E21:32"/><i><font color="Green" title="E43">A17</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1d.html#E5:32"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan1d.html#E8:32"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="jordan1d.html#E9:32"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan1d.html#E17:32"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="jordan1d.html#E19:32"><i><font color="Green" title="E41">A15</font></i></a>, <a class="ref" href="jordan1a.html#T102" title="JORDAN1A:th.102">JORDAN1A:102</a>, <a class="ref" href="sppol_1.html#T64" title="SPPOL_1:th.64">SPPOL_1:64</a></i>;<br/>
<a NAME="E22:32"/><i><font color="Green" title="E44">A18</font></i>: 
 <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="c2">C</font>,<font color="Maroon" title="c1">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="c2">C</font>,<font color="Maroon" title="c1">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="E23:32"/><i><font color="Green" title="E45">A19</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i1</font> &amp; <font color="Maroon" title="c5">i1</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1d.html#E12:32"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E24:32"/><i><font color="Green" title="E46">A20</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i2</font> &amp; <font color="Maroon" title="c7">i2</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1d.html#E14:32"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E25:32"/><i><font color="Green" title="E47">A21</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j1</font> &amp; <font color="Maroon" title="c8">j2</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1d.html#E12:32"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan1d.html#E14:32"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E26:32"/><i><font color="Green" title="E48">A22</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j2</font> &amp; <font color="Maroon" title="c6">j1</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1d.html#E12:32"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan1d.html#E14:32"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E27:32"/><i><font color="Green" title="E49">A23</font></i>: 
<font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c5">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i2</font>
 ;<br/>

<a NAME="E2:32_1"/><b>then </b>
( <font color="Maroon" title="c5">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">i2</font> or <font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">i1</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:32_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1d.html#E13:32"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan1d.html#E15:32"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="jordan1d.html#E16:32"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="jordan1d.html#E20:32"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="jordan1d.html#E21:32"><i><font color="Green" title="E43">A17</font></i></a>, <a class="txt" href="jordan1d.html#E23:32"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan1d.html#E24:32"><i><font color="Green" title="E46">A20</font></i></a>, <a class="txt" href="jordan1d.html#E25:32"><i><font color="Green" title="E47">A21</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:32"/>
<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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">j1</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"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1d.html#E13:32"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan1d.html#E18:32"><i><font color="Green" title="E40">A14</font></i></a>, <a class="txt" href="jordan1d.html#E20:32"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="jordan1d.html#E22:32"><i><font color="Green" title="E44">A18</font></i></a>, <a class="txt" href="jordan1d.html#E25:32"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan1d.html#E26:32"><i><font color="Green" title="E48">A22</font></i></a>, <a class="ref" href="jordan1a.html#T94" title="JORDAN1A:th.94">JORDAN1A:94</a></i>;<br/>
<a NAME="E29:32"/><b>then </b><i><font color="Green" title="E50">A24</font></i>: 
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">i1</font>
 
<b>by </b><i><a class="txt" href="jordan1d.html#E23:32"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan1d.html#E25:32"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan1d.html#E26:32"><i><font color="Green" title="E48">A22</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/>
<a NAME="E30:32"/><b>then </b><i><font color="Green" title="E51">A25</font></i>: 
<font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan1d.html#E23:32"><i><font color="Green" title="E45">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E31:32"/><i><font color="Green" title="E52">A26</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j1</font> &amp; <font color="Maroon" title="c6">j1</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan1d.html#E7:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="jordan1d.html#E12:32"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan1d.html#E13:32"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan1d.html#E14:32"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="jordan1d.html#E15:32"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="jordan1d.html#E16:32"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="jordan1d.html#E25:32"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan1d.html#E27:32"><i><font color="Green" title="E49">A23</font></i></a>, <a class="txt" href="jordan1d.html#E29:32"><i><font color="Green" title="E50">A24</font></i></a>, <a class="ref" href="jordan10.html#T2" title="JORDAN10:th.2">JORDAN10:2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E32:32"/><i><font color="Green" title="E53">A27</font></i>: 
 <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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 
<b>by </b><i><a class="ref" href="jordan8.html#T13" title="JORDAN8:th.13">JORDAN8:13</a></i>;<br/>
<a NAME="E33:32"/><b>then </b><i><font color="Green" title="E54">A28</font></i>: 
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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:32"/><i><font color="Green" title="E55">A29</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1: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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1d.html#E32:32"><i><font color="Green" title="E53">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<i><font color="Green" title="E56">A30</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
 <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c2">C</font>
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1d.html#E33:32"><i><font color="Green" title="E54">A28</font></i></a>, <a class="ref" href="jordan8.html#T14" title="JORDAN8:th.14">JORDAN8:14</a></i>
;<br/>
<a NAME="E36:32"/><b>then </b><i><font color="Green" title="E57">A31</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i1</font>,1</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1d.html#E22:32"><i><font color="Green" title="E44">A18</font></i></a>, <a class="txt" href="jordan1d.html#E30:32"><i><font color="Green" title="E51">A25</font></i></a>, <a class="txt" href="jordan1d.html#E33:32"><i><font color="Green" title="E54">A28</font></i></a>, <a class="txt" href="jordan1d.html#E34:32"><i><font color="Green" title="E55">A29</font></i></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></i>;<br/>
<a NAME="E37:32"/>
<font color="Maroon" title="c6">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="E38:32"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="jordan1d.html#E7:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="jordan1d.html#E12:32"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan1d.html#E13:32"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan1d.html#E14:32"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="jordan1d.html#E15:32"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="jordan1d.html#E16:32"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="jordan1d.html#E23:32"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan1d.html#E25:32"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan1d.html#E27:32"><i><font color="Green" title="E49">A23</font></i></a>, <a class="txt" href="jordan1d.html#E29:32"><i><font color="Green" title="E50">A24</font></i></a>, <a class="ref" href="jordan10.html#T2" title="JORDAN10:th.2">JORDAN10:2</a>, <a class="ref" href="jordan1a.html#T40" title="JORDAN1A:th.40">JORDAN1A:40</a></i>;<br/>
<a NAME="E39:32"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</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="c3">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c3">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="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="jordan1d.html#E8:32"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="jordan1d.html#E19:32"><i><font color="Green" title="E41">A15</font></i></a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/>
<a NAME="E40:32"/><b>then </b><i><font color="Green" title="E58">A32</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j1</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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</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> <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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="jordan1d.html#E5:32"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan1d.html#E7:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="jordan1d.html#E12:32"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan1d.html#E13:32"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan1d.html#E14:32"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="jordan1d.html#E15:32"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="jordan1d.html#E16:32"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="jordan1d.html#E27:32"><i><font color="Green" title="E49">A23</font></i></a>, <a class="txt" href="jordan1d.html#E30:32"><i><font color="Green" title="E51">A25</font></i></a>, <a class="ref" href="jordan10.html#T2" title="JORDAN10:th.2">JORDAN10:2</a>, <a class="ref" href="topreal1.html#D15" title="TOPREAL1:def.15">TOPREAL1:def 15</a></i>;<br/>
<a NAME="E41:32"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</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#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<a NAME="E42:32"/><b>then </b>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i1</font>,1</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="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <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="c2">C</font>,<font color="Maroon" title="c1">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="c5">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j1</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="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan1d.html#E30:32"><i><font color="Green" title="E51">A25</font></i></a>, <a class="txt" href="jordan1d.html#E35:32"><i><font color="Green" title="E56">A30</font></i></a>, <a class="txt" href="jordan1d.html#E36:32"><i><font color="Green" title="E57">A31</font></i></a>, <a class="txt" href="jordan1d.html#E40:32"><i><font color="Green" title="E58">A32</font></i></a></i>;<br/>
<a NAME="E43:32"/><b>then </b>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</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="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">i1</font>,<font color="Maroon" title="c6">j1</font>
 
<b>by </b><i><a class="txt" href="jordan1d.html#E30:32"><i><font color="Green" title="E51">A25</font></i></a>, <a class="txt" href="jordan1d.html#E31:32"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="jordan1d.html#E33:32"><i><font color="Green" title="E54">A28</font></i></a>, <a class="ref" href="gobrd11.html#T32" title="GOBRD11:th.32">GOBRD11:32</a></i>;<br/>
<b>hence </b><a NAME="E44:32"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1d.html#E7:32"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="jordan1d.html#E10:32"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="jordan1d.html#E12:32"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan1d.html#E13:32"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan1d.html#E14:32"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="jordan1d.html#E15:32"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="jordan1d.html#E16:32"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="jordan1d.html#E27:32"><i><font color="Green" title="E49">A23</font></i></a>, <a class="txt" href="jordan1d.html#E29:32"><i><font color="Green" title="E50">A24</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a>, <a class="ref" href="jordan10.html#T2" title="JORDAN10:th.2">JORDAN10:2</a></i>;<br/>


</div>
