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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">E</font> be  <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>









<b>set </b><font color="Maroon">a</font> =  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">E</font>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">E</font>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">m</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">m</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:57"/>
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">E</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T34">SPRECT_1:34</a></i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green">E71</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">E</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">E</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E5:57"/><i><font color="Green">E75</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:57"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E7:57"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E8:57"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">E</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">E</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">E</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">E</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">E</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">E</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:57"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , </i>;<br/>


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