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<b>let </b><font color="Maroon">h</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><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">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">i1</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">Y</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:31"/><i><font color="Green">E38</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> <font color="Maroon">h</font> &amp; 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="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">j</font>,<font color="Maroon">I</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> &amp;  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">j</font>,<font color="Maroon">I</font> ) ) </span>}</span>  &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">I</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:31"/><b>then </b>
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:31"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">I</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> &amp;  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">I</font> ) )
 <b>by </b><i><a class="txt" href="jordan5d.html#E9:31"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E9:31"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:31"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="goboard5.html#T12">GOBOARD5:12</a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</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="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan5d.html#E10:31"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E9:31"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">I</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan5d.html#E9:31"><i><font color="Green">E44</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/>
<i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan5d.html#E10:31"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan5d.html#E12:31"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">I</font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="jordan5d.html#E9:31"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan5d.html#E12:31"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>
;<br/>
<a NAME="E11:31"/>
<font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E12:31"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">I</font>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E15:31"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan5d.html#E16:31"><i><font color="Green">E50</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E13:31"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">I</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E9:31"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan5d.html#E12:31"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E14:31"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E9:31"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="jordan5d.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E15:31"/><b>then </b><i><font color="Green">E49</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</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="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan5d.html#E9:31"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan5d.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan5d.html#E10:31"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a>, <a class="ref" href="goboard7.html#T35">GOBOARD7:35</a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E16:31"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i1</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="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan5d.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1"/>
( <font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font> or <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1"><b>case </b></a><a NAME="E1:31_2_1_1"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">I</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">I</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E9:31"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan5d.html#E12:31"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan5d.html#T8" target="_self">Th8</a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2"><b>case </b></a><a NAME="E1:31_2_1_2"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">I</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">I</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="jordan5d.html#T7" target="_self">Th7</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E18:31"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">h</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">I</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E16:31"><i><font color="Green">E50</font></i></a></i>;<br/>


</div>
