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<b>let </b><font color="Maroon" title="c1">j</font>, <font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">G</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="goboard1.html#V3" title="GOBOARD1:attr.3">X_equal-in-line</a>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green" title="E7">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">j</font> &amp; <font color="Maroon" title="c1">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="c3">G</font> )
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green" title="E8">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">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> )
 ;<br/>

<a NAME="E4:4"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E3:4"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:4"/><b>then </b><i><font color="Green" title="E9">A4</font></i>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">i</font></span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:4"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="goboard1.html#D6" title="GOBOARD1:def.6">GOBOARD1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">c</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">i</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E7:4"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:4"/><i><font color="Green" title="E11">A6</font></i>: 
1 <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="c3">G</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:4"/><b>then </b><i><font color="Green" title="E12">A7</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:4"/><i><font color="Green" title="E13">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E11:4"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E8:4"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E12:4"/><b>then </b><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E13:4"/>
<font color="Maroon" title="c1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="goboard5.html#E10:4"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:4"/><b>then </b><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E15:4"/><i><font color="Green" title="E16">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E16:4"/><b>then </b><i><font color="Green" title="E17">A12</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E8:4"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="goboard5.html#E10:4"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:4"/><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon" title="c1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:4"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="goboard5.html#E10:4"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="goboard5.html#E15:4"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E7:4"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="goboard5.html#E14:4"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c1">j</font>
<b>by </b><i><a class="txt" href="goboard5.html#E17:4"><i><font color="Green" title="E18">A13</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="goboard5.html#E5:4"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="goboard5.html#E16:4"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="goboard5.html#E17:4"><i><font color="Green" title="E18">A13</font></i></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E16:4"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E9:4"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="goboard5.html#E12:4"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
;<br/>


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