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<span class="kw">let </span><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">j1</font>, <font color="Maroon" title="c3">i2</font>, <font color="Maroon" title="c4">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being  <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>  st  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Olive" title="b1">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Olive" title="b2">G2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> &amp; <font color="Maroon" title="c1">i1</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">G1</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font> &amp; <font color="Maroon" title="c2">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">G1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font> &amp; <font color="Maroon" title="c3">i2</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="b2">G2</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font> &amp; <font color="Maroon" title="c4">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b2">G2</font> &amp; <font color="Olive" title="b1">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">j2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c5">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c6">G2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> &amp; <font color="Maroon" title="c1">i1</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="c5">G1</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font> &amp; <font color="Maroon" title="c2">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font> &amp; <font color="Maroon" title="c3">i2</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="c6">G2</font> &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font> &amp; <font color="Maroon" title="c4">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">G2</font> &amp; <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">j2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
 <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c5">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c6">G2</font>
 <span class="kw">and </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> &amp; <font color="Maroon" title="c1">i1</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="c5">G1</font> )
 <span class="kw">and </span><br/><a NAME="E3:20"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font>
 <span class="kw">and </span><br/><a NAME="E4:20"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c2">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E5:20"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font> &amp; <font color="Maroon" title="c3">i2</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="c6">G2</font> )
 <span class="kw">and </span><br/><a NAME="E6:20"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font>
 <span class="kw">and </span><br/><a NAME="E7:20"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c4">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">G2</font>
 <span class="kw">and </span><br/><a NAME="E8:20"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<a NAME="E9:20"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/>
<a NAME="E10:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T51" title="NAT_D:th.51">NAT_D:51</a></span>;<br/>
<a NAME="E11:20"/><span class="kw">then </span>
<font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:20"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="matrix_1.html#T37" title="MATRIX_1:th.37">MATRIX_1:37</a></span>;<br/>
<a NAME="E13:20"/><span class="kw">then </span>
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> </span>}</span> 
 
;<br/>
<a NAME="E14:20"/><span class="kw">then </span>
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#T7" target="_self" title="GOBRD13:th.7">Th7</a></span>;<br/>
<a NAME="E15:20"/><span class="kw">then </span>
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c6">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<a NAME="E16:20"/><span class="kw">then </span>
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">G2</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#T7" target="_self" title="GOBRD13:th.7">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E18:20"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>
 <span class="kw">and </span><br/><a NAME="E19:20"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c6">G2</font>
 ;<br/>
<a NAME="E20:20"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E21:20"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<a NAME="E22:20"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">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="c6">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E29">A15</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">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="#E20:20"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c9">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="#E5:20"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></span>
;<br/>
<a NAME="E24:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">j2</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="#E2:20"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E18:20"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E31">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_2"/>
<font color="Maroon" title="c4">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_2"/><span class="kw">then </span>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j2</font> or <font color="Maroon" title="c4">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E21:20"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E24:20"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E26:20"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/>
<a NAME="E27:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T51" title="NAT_D:th.51">NAT_D:51</a></span>;<br/>
<a NAME="E28:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E29:20"/><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_3"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:20_3"/><span class="kw">then </span>
( <font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> or <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:20_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E29:20"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E18:20"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E23:20"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E28:20"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E31:20"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:20"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
