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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">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> <font color="Olive" title="b1">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b1">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,<font color="Maroon" title="c2">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font></span>)</span>,<font color="Maroon" title="c2">j</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">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> <font color="Maroon" title="c3">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,<font color="Maroon" title="c2">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c2">j</font>)</span>)</span>) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">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> <font color="Maroon" title="c3">G</font> )
 <span class="kw">and </span><br/><a NAME="E2:19"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c3">G</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,<font color="Maroon" title="c2">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c2">j</font>)</span>)</span>)</span><br/>

<a NAME="E3:19"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="sprect_3.html#T13" onmouseover="rs('sprect_3/T13')" onmouseout="rh()">SPRECT_3:13</a></span>;<br/>
<a NAME="E4:19"/>
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> <font color="Maroon" title="c3">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="goboard5.html#T1" onmouseover="rs('goboard5/T1')" onmouseout="rh()">GOBOARD5:1</a></span>;<br/>
<a NAME="E6:19"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="goboard5.html#T1" onmouseover="rs('goboard5/T1')" onmouseout="rh()">GOBOARD5:1</a>, <a class="ref" href="sprect_3.html#T13" onmouseover="rs('sprect_3/T13')" onmouseout="rh()">SPRECT_3:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:19"/>
<font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,<font color="Maroon" title="c2">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c2">j</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="goboard7.html#T8" onmouseover="rs('goboard7/T8')" onmouseout="rh()">GOBOARD7:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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