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<div class="add">

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">bD</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">MD</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">D</font>;<br/>





<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E28">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">MD</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">bD</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:30"/>
( <font color="Maroon" title="c3">MD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c2">bD</font> implies (  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">MD</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">bD</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">MD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c3">MD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c2">bD</font>
 ;<br/>

<a NAME="E2:30_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <font color="Maroon" title="c2">bD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrix15.html#T25" target="_self" title="MATRIX15:th.25">Th25</a></i>;<br/>
<a NAME="E3:30_1"/><b>then </b>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <font color="Maroon" title="c2">bD</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp; 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> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b> <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">MD</font>,1 = 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <font color="Maroon" title="c2">bD</font></span>)</span>,1
<b>by </b><i><a class="txt" href="matrix15.html#E1:30_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="matrix_2.html#T16" title="MATRIX_2:th.16">MATRIX_2:16</a></i>
<br/>.= 
<font color="Maroon" title="c2">bD</font>
<b>by </b><i><a class="ref" href="matrix15.html#T25" target="_self" title="MATRIX15:th.25">Th25</a></i>
;<br/><br/>

<a NAME="E5:30_1"/><i><font color="Green" title="E30">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">MD</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="c2">bD</font>
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:30_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E6:30_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">MD</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:30"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:30_1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">MD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="matrix15.html#E1:30_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="matrix15.html#E5:30_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:30"/><i><font color="Green" title="E29">A4</font></i>: 
(  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c3">MD</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">bD</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">MD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>

<a NAME="E4:30"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">bD</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="c3">MD</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">MD</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E5:30"/><b>then </b><i><font color="Green" title="E31">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">MD</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:30"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E6:30"/>
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> 1
 
;<br/>
<a NAME="E7:30"/><b>then </b>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">MD</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">bD</font>
 
<b>by </b><i><a class="txt" href="matrix15.html#E3:30"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="matrix_2.html#T17" title="MATRIX_2:th.17">MATRIX_2:17</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <font color="Maroon" title="c2">bD</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">MD</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
<b>by </b><i><a class="txt" href="matrix15.html#E4:30"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="matrix15.html#T25" target="_self" title="MATRIX15:th.25">Th25</a></i>
<br/>.= 
<font color="Maroon" title="c3">MD</font>
<b>by </b><i><a class="txt" href="matrix15.html#E3:30"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="matrix15.html#E5:30"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="matrix_2.html#T15" title="MATRIX_2:th.15">MATRIX_2:15</a></i>
;<br/>
<b>hence </b><a NAME="E9:30"/>
<font color="Maroon" title="c3">MD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c2">bD</font>
 ;<br/>


</div>
