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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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>let </b><font color="Maroon">c<sub>4</sub></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>assume </b><a NAME="E1:35"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font></span>)</span> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Olive">b<sub>1</sub></font> ) ) </span>}</span>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</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">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="jordan5d.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green">E50</font></i>: 
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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard7.html#T35">GOBOARD7:35</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:35"/><i><font color="Green">E51</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan5d.html#E1:35"><i><font color="Green">E48</font></i></a>, <a class="ref" href="sppol_2.html#T14">SPPOL_2:14</a></i>;<br/>
<a NAME="E6:35"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:35"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:35"/><i><font color="Green">E53</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="sppol_1.html#T5">SPPOL_1:5</a></i>;<br/>
<a NAME="E9:35"/><i><font color="Green">E54</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="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard7.html#T34">GOBOARD7:34</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1"/>
(  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> is <a href="sppol_1.html#V2">vertical</a> or  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> is <a href="sppol_1.html#V1">horizontal</a> )
 </span><b>by </b><i><a class="ref" href="sppol_1.html#T41">SPPOL_1:41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1"><b>case </b></a><a NAME="E1:35_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> is <a href="sppol_1.html#V2">vertical</a>
 ;<br/><div class="add"><a NAME="E2:35_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is <a href="sppol_1.html#V2">vertical</a>
 <b>by </b><i><a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:35_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/><a NAME="E4:35_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:35_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:35_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E3:35_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E1:35"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan5d.html#E2:35"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan5d.html#E7:35"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan5d.html#E9:35"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan5d.html#E4:35_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan5d.html#T43" target="_self">Th43</a></i>;<br/><b>hence </b><a NAME="E4:35_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E2:35_1_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:35_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:35_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E3:35_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E1:35"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan5d.html#E2:35"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan5d.html#E8:35"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan5d.html#E9:35"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan5d.html#E4:35_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan5d.html#T43" target="_self">Th43</a></i>;<br/><b>hence </b><a NAME="E4:35_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E2:35_1_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:35_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2"><b>case </b></a><a NAME="E1:35_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> is <a href="sppol_1.html#V1">horizontal</a>
 ;<br/><div class="add"><a NAME="E2:35_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is <a href="sppol_1.html#V1">horizontal</a>
 <b>by </b><i><a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:35_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a></i>;<br/><a NAME="E4:35_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard7.html#T6">GOBOARD7:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:35_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:35_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E2:35"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><a NAME="E3:35_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E3:35"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan5d.html#E7:35"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5d.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E4:35_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</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">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E2:35_1_1_2_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E1:35"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan5d.html#E7:35"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan5d.html#E9:35"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan5d.html#E4:35_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan5d.html#T43" target="_self">Th43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:35_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:35_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E2:35"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/><a NAME="E3:35_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E3:35"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan5d.html#E8:35"><i><font color="Green">E53</font></i></a>, <a class="ref" href="jordan5d.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E4:35_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</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">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E2:35_1_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan5d.html#E1:35"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan5d.html#E5:35"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan5d.html#E8:35"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan5d.html#E9:35"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan5d.html#E4:35_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan5d.html#T43" target="_self">Th43</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:35_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:35"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
 ;<br/>


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