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<b>let </b><font color="Maroon" title="c1">j</font>, <font color="Maroon" title="c2">i1</font>, <font color="Maroon" title="c3">i2</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> ;<br/><b>let </b><font color="Maroon" title="c4">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="E1:5"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c4">G</font> is <a href="goboard1.html#V6" title="GOBOARD1:attr.6">X_increasing-in-column</a>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green" title="E8">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="c4">G</font> )
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green" title="E9">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> &amp; <font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c4">G</font> )
 ;<br/>

<a NAME="E4:5"/>
<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="c4">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:5"><i><font color="Green" title="E8">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="E5:5"/><b>then </b><i><font color="Green" title="E10">A4</font></i>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c1">j</font></span>)</span> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:5"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">c</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c1">j</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="E7:5"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c2">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="c4">G</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E3:5"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:5"/><b>then </b><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c2">i1</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">G</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E3:5"><i><font color="Green" title="E9">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="E9:5"/><i><font color="Green" title="E13">A7</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E3:5"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:5"/><b>then </b><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c3">i2</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">G</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E3:5"><i><font color="Green" title="E9">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="E11:5"/><i><font color="Green" title="E15">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">c</font> <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">G</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E12:5"/><b>then </b>
<font color="Maroon" title="c2">i1</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="c5">c</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E3:5"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="goboard5.html#E7:5"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:5"/><b>then </b><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i1</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E14:5"/>
<font color="Maroon" title="c3">i2</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="c5">c</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E3:5"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="goboard5.html#E9:5"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="goboard5.html#E11:5"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:5"/><b>then </b><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E16:5"/><i><font color="Green" title="E18">A12</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#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c1">j</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="c5">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="E17:5"/><b>then </b><i><font color="Green" title="E19">A13</font></i>: 
<font color="Maroon" title="c2">i1</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#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c1">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E3:5"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="goboard5.html#E7:5"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="goboard5.html#E11:5"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green" title="E20">A14</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</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="c5">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">i1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E8:5"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="goboard5.html#E13:5"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c1">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i1</font>
<b>by </b><i><a class="txt" href="goboard5.html#E17:5"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>
;<br/>
<a NAME="E19:5"/><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c3">i2</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#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c1">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E3:5"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="goboard5.html#E9:5"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="goboard5.html#E11:5"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="goboard5.html#E16:5"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c1">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><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="c4">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<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#E10:5"><i><font color="Green" title="E14">A8</font></i></a>, <a class="txt" href="goboard5.html#E15:5"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E21:5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<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#E3:5"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="goboard5.html#E5:5"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="goboard5.html#E17:5"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="goboard5.html#E18:5"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="goboard5.html#E19:5"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>


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