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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM8">FinSequence</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>2</sub></font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green">E3</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 ;<br/>

<a NAME="E3:3"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V3">s.n.c.</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V1">special</a> )
 
<b>by </b><i><a class="txt" href="goboard3.html#E2:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:3"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:3"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="topreal1.html#V3">s.n.c.</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="topreal1.html#V1">special</a> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="ref" href="goboard3.html#T1" target="_self">Th1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E7:3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard3.html#E5:3"><i><font color="Green">E6</font></i></a></i>;<br/>

<a NAME="E8:3"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="goboard3.html#E3:3"><i><font color="Green">E5</font></i></a>, <a class="txt" href="goboard3.html#E6:3"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <b>by </b><i><a class="txt" href="goboard3.html#E6:3"><i><font color="Green">E7</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>

<b>thus </b><a NAME="E10:3"/>
(  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard3.html#E6:3"><i><font color="Green">E7</font></i></a></i>;<br/>


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