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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">E</font> be  <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/>









<b>set </b><font color="Maroon" title="c6">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">m</font>;<br/>
<b>set </b><font color="Maroon" title="c7">G1</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c2">n</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green" title="E39">A2</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">i</font>
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c3">i</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="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="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span>
 <b>and </b><br/><a NAME="E4:61"/><i><font color="Green" title="E41">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">j</font>
 <b>and </b><br/><a NAME="E5:61"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c4">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="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span>
 ;<br/>

<a NAME="E6:61"/>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">i</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="E7:61"/><b>then </b><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon" title="c3">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="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E3:61"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:61"/>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">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="E9:61"/><b>then </b><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c4">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="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E5:61"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:61"/><i><font color="Green" title="E45">A8</font></i>: 
1 <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> <font color="Maroon" title="c3">i</font>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E11:61"/><i><font color="Green" title="E46">A9</font></i>: 
1 <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> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">i1</font>, <font color="Maroon" title="c9">j1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E12:61"/><i><font color="Green" title="E47">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <b>and </b><br/><a NAME="E13:61"/><i><font color="Green" title="E48">A11</font></i>: 
<font color="Maroon" title="c8">i1</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">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2
 <b>and </b><br/><a NAME="E14:61"/><i><font color="Green" title="E49">A12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 <b>and </b><br/><a NAME="E15:61"/><i><font color="Green" title="E50">A13</font></i>: 
<font color="Maroon" title="c9">j1</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">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2
 ;<br/>

<b>let </b><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E16:61"/><i><font color="Green" title="E51">A14</font></i>: 
<font color="Maroon" title="c10">e</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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">p</font> = <font color="Maroon" title="c10">e</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#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c12">i2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<b>set </b><font color="Maroon" title="c13">j2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<b>set </b><font color="Maroon" title="c14">i3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<b>set </b><font color="Maroon" title="c15">j3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<a NAME="E18:61"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 <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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan1a.html#E3:61"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="jordan1a.html#T55" target="_self" title="JORDAN1A:th.55">Th55</a></i>;<br/>
<a NAME="E19:61"/><b>then </b><i><font color="Green" title="E52">A15</font></i>: 
<font color="Maroon" title="c8">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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E13:61"><i><font color="Green" title="E48">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:61"/><b>then </b><i><font color="Green" title="E53">A16</font></i>: 
<font color="Maroon" title="c8">i1</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="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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E21:61"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 <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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan1a.html#E5:61"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="jordan1a.html#T56" target="_self" title="JORDAN1A:th.56">Th56</a></i>;<br/>
<a NAME="E22:61"/><b>then </b><i><font color="Green" title="E54">A17</font></i>: 
<font color="Maroon" title="c9">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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E15:61"><i><font color="Green" title="E50">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:61"/><b>then </b><i><font color="Green" title="E55">A18</font></i>: 
<font color="Maroon" title="c9">j1</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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E24:61"/><i><font color="Green" title="E56">A19</font></i>: 
1 <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">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan1a.html#E7:61"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="jordan1a.html#T52" target="_self" title="JORDAN1A:th.52">Th52</a></i>;<br/>
<a NAME="E25:61"/><b>then </b><i><font color="Green" title="E57">A20</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E12:61"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E26:61"/><i><font color="Green" title="E58">A21</font></i>: 
1 <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">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan1a.html#E9:61"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="jordan1a.html#T53" target="_self" title="JORDAN1A:th.53">Th53</a></i>;<br/>
<a NAME="E27:61"/><b>then </b><i><font color="Green" title="E59">A22</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E14:61"><i><font color="Green" title="E49">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E28:61"/><b>then </b><i><font color="Green" title="E60">A23</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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</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="c11">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c11">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="c5">E</font>,<font color="Maroon" title="c2">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="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font></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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">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="Maroon" title="c11">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c11">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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</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="jordan1a.html#E16:61"><i><font color="Green" title="E51">A14</font></i></a>, <a class="txt" href="jordan1a.html#E20:61"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="jordan1a.html#E23:61"><i><font color="Green" title="E55">A18</font></i></a>, <a class="txt" href="jordan1a.html#E25:61"><i><font color="Green" title="E57">A20</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></i>;<br/>
<a NAME="E29:61"/><i><font color="Green" title="E61">A24</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1a.html#E10:61"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="jordan1a.html#E11:61"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E30:61"/><b>then </b><i><font color="Green" title="E62">A25</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan1a.html#E7:61"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="jordan1a.html#E9:61"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="jordan1a.html#T54" target="_self" title="JORDAN1A:th.54">Th54</a></i>;<br/>
<a NAME="E31:61"/><i><font color="Green" title="E63">A26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E12:61"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="jordan1a.html#E19:61"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan1a.html#E29:61"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:61"/><i><font color="Green" title="E64">A27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E14:61"><i><font color="Green" title="E49">A12</font></i></a>, <a class="txt" href="jordan1a.html#E22:61"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan1a.html#E29:61"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E33:61"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">i1</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> )
 
<b>by </b><i><a class="txt" href="jordan1a.html#E12:61"><i><font color="Green" title="E47">A10</font></i></a>, <a class="txt" href="jordan1a.html#E29:61"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E34:61"/><b>then </b><i><font color="Green" title="E65">A28</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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c9">j1</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> <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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1a.html#E19:61"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan1a.html#E22:61"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan1a.html#E24:61"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="jordan1a.html#E27:61"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="jordan1a.html#E29:61"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1a.html#E22:61"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan1a.html#E24:61"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="jordan1a.html#E27:61"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="jordan1a.html#E29:61"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan1a.html#E31:61"><i><font color="Green" title="E63">A26</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1a.html#E24:61"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="jordan1a.html#E26:61"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="jordan1a.html#E29:61"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan1a.html#E31:61"><i><font color="Green" title="E63">A26</font></i></a>, <a class="txt" href="jordan1a.html#E32:61"><i><font color="Green" title="E64">A27</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>
;<br/>
<a NAME="E36:61"/><b>then </b><i><font color="Green" title="E66">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="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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="c11">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1a.html#E28:61"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan1a.html#E30:61"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan1a.html#E34:61"><i><font color="Green" title="E65">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E37:61"/><i><font color="Green" title="E67">A30</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E38:61"/><i><font color="Green" title="E68">A31</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E39:61"/><i><font color="Green" title="E69">A32</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan1a.html#E25:61"><i><font color="Green" title="E57">A20</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E40:61"/><i><font color="Green" title="E70">A33</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan1a.html#E27:61"><i><font color="Green" title="E59">A22</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<i><font color="Green" title="E71">A34</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">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> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1

<br/>.= 
<font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/>
<i><font color="Green" title="E72">A35</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">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> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c4">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1

<br/>.= 
<font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/>
<a NAME="E43:61"/><i><font color="Green" title="E73">A36</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1a.html#E10:61"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="jordan1a.html#E11:61"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E44:61"/><b>then </b><i><font color="Green" title="E74">A37</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">m</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="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">j</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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E3:61"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan1a.html#E9:61"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="jordan1a.html#E37:61"><i><font color="Green" title="E67">A30</font></i></a>, <a class="txt" href="jordan1a.html#E41:61"><i><font color="Green" title="E71">A34</font></i></a>, <a class="ref" href="jordan1a.html#T54" target="_self" title="JORDAN1A:th.54">Th54</a></i>;<br/>
<a NAME="E45:61"/><i><font color="Green" title="E75">A38</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E46:61"/><b>then </b><i><font color="Green" title="E76">A39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan1a.html#E3:61"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="jordan1a.html#T55" target="_self" title="JORDAN1A:th.55">Th55</a></i>;<br/>
<a NAME="E47:61"/>
<font color="Maroon" title="c8">i1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan1a.html#E13:61"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="jordan1a.html#E45:61"><i><font color="Green" title="E75">A38</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E48:61"/><b>then </b>
( <font color="Maroon" title="c8">i1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E49:61"/><b>then </b><i><font color="Green" title="E77">A40</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="c5">E</font>,<font color="Maroon" title="c2">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="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</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> <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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1a.html#E22:61"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan1a.html#E27:61"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="jordan1a.html#E39:61"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="jordan1a.html#E46:61"><i><font color="Green" title="E76">A39</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/>
<a NAME="E50:61"/><i><font color="Green" title="E78">A41</font></i>: 
1 <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">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan1a.html#E13:61"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="jordan1a.html#E25:61"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="jordan1a.html#E45:61"><i><font color="Green" title="E75">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1a.html#E22:61"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan1a.html#E27:61"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="jordan1a.html#E46:61"><i><font color="Green" title="E76">A39</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1a.html#E26:61"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="jordan1a.html#E32:61"><i><font color="Green" title="E64">A27</font></i></a>, <a class="txt" href="jordan1a.html#E43:61"><i><font color="Green" title="E73">A36</font></i></a>, <a class="txt" href="jordan1a.html#E46:61"><i><font color="Green" title="E76">A39</font></i></a>, <a class="txt" href="jordan1a.html#E50:61"><i><font color="Green" title="E78">A41</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>
;<br/>
<a NAME="E52:61"/><b>then </b><i><font color="Green" title="E79">A42</font></i>: 
<font color="Maroon" title="c11">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="c5">E</font>,<font color="Maroon" title="c1">m</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="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1a.html#E28:61"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan1a.html#E44:61"><i><font color="Green" title="E74">A37</font></i></a>, <a class="txt" href="jordan1a.html#E49:61"><i><font color="Green" title="E77">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E53:61"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">j1</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> )
 
<b>by </b><i><a class="txt" href="jordan1a.html#E14:61"><i><font color="Green" title="E49">A12</font></i></a>, <a class="txt" href="jordan1a.html#E43:61"><i><font color="Green" title="E73">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E54:61"/><b>then </b><i><font color="Green" title="E80">A43</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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1a.html#E19:61"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan1a.html#E22:61"><i><font color="Green" title="E54">A17</font></i></a>, <a class="txt" href="jordan1a.html#E25:61"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="jordan1a.html#E26:61"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="jordan1a.html#E43:61"><i><font color="Green" title="E73">A36</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="txt" href="jordan1a.html#E19:61"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan1a.html#E25:61"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="jordan1a.html#E26:61"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="jordan1a.html#E32:61"><i><font color="Green" title="E64">A27</font></i></a>, <a class="txt" href="jordan1a.html#E43:61"><i><font color="Green" title="E73">A36</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="txt" href="jordan1a.html#E24:61"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="jordan1a.html#E26:61"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="jordan1a.html#E31:61"><i><font color="Green" title="E63">A26</font></i></a>, <a class="txt" href="jordan1a.html#E32:61"><i><font color="Green" title="E64">A27</font></i></a>, <a class="txt" href="jordan1a.html#E43:61"><i><font color="Green" title="E73">A36</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>
;<br/>
<a NAME="E56:61"/><b>then </b><i><font color="Green" title="E81">A44</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="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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="c11">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1a.html#E28:61"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan1a.html#E30:61"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan1a.html#E54:61"><i><font color="Green" title="E80">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E57:61"/><i><font color="Green" title="E82">A45</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 )
 
<b>by </b><i><a class="txt" href="jordan1a.html#E10:61"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="jordan1a.html#E11:61"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E58:61"/><b>then </b><i><font color="Green" title="E83">A46</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">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">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p3">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan1a.html#E5:61"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="jordan1a.html#E7:61"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="jordan1a.html#E38:61"><i><font color="Green" title="E68">A31</font></i></a>, <a class="txt" href="jordan1a.html#E42:61"><i><font color="Green" title="E72">A35</font></i></a>, <a class="ref" href="jordan1a.html#T54" target="_self" title="JORDAN1A:th.54">Th54</a></i>;<br/>
<a NAME="E59:61"/><i><font color="Green" title="E84">A47</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E60:61"/><b>then </b><i><font color="Green" title="E85">A48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:61"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan1a.html#E5:61"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="jordan1a.html#T56" target="_self" title="JORDAN1A:th.56">Th56</a></i>;<br/>
<a NAME="E61:61"/>
<font color="Maroon" title="c9">j1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan1a.html#E15:61"><i><font color="Green" title="E50">A13</font></i></a>, <a class="txt" href="jordan1a.html#E59:61"><i><font color="Green" title="E84">A47</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E62:61"/><b>then </b>
( <font color="Maroon" title="c9">j1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E63:61"/><b>then </b><i><font color="Green" title="E86">A49</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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">j1</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="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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1a.html#E19:61"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan1a.html#E25:61"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="jordan1a.html#E40:61"><i><font color="Green" title="E70">A33</font></i></a>, <a class="txt" href="jordan1a.html#E60:61"><i><font color="Green" title="E85">A48</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>;<br/>
<a NAME="E64:61"/><i><font color="Green" title="E87">A50</font></i>: 
1 <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">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan1a.html#E15:61"><i><font color="Green" title="E50">A13</font></i></a>, <a class="txt" href="jordan1a.html#E27:61"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="jordan1a.html#E59:61"><i><font color="Green" title="E84">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</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="c5">E</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="txt" href="jordan1a.html#E19:61"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="jordan1a.html#E25:61"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="jordan1a.html#E60:61"><i><font color="Green" title="E85">A48</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</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="c5">E</font>,<font color="Maroon" title="c2">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"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="txt" href="jordan1a.html#E24:61"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="jordan1a.html#E31:61"><i><font color="Green" title="E63">A26</font></i></a>, <a class="txt" href="jordan1a.html#E57:61"><i><font color="Green" title="E82">A45</font></i></a>, <a class="txt" href="jordan1a.html#E60:61"><i><font color="Green" title="E85">A48</font></i></a>, <a class="txt" href="jordan1a.html#E64:61"><i><font color="Green" title="E87">A50</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>
;<br/>
<a NAME="E66:61"/><b>then </b>
<font color="Maroon" title="c11">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="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">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="jordan1a.html#E28:61"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="jordan1a.html#E58:61"><i><font color="Green" title="E83">A46</font></i></a>, <a class="txt" href="jordan1a.html#E63:61"><i><font color="Green" title="E86">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E67:61"/>
<font color="Maroon" title="c10">e</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="c5">E</font>,<font color="Maroon" title="c1">m</font></span>)</span>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="jordan1a.html#E2:61"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan1a.html#E3:61"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="jordan1a.html#E4:61"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan1a.html#E5:61"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="jordan1a.html#E36:61"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan1a.html#E52:61"><i><font color="Green" title="E79">A42</font></i></a>, <a class="txt" href="jordan1a.html#E56:61"><i><font color="Green" title="E81">A44</font></i></a>, <a class="ref" href="jordan9.html#T19" title="JORDAN9:th.19">JORDAN9:19</a></i>;<br/>


</div>
