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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c3">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c1">n</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>  &amp; not  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:64"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">X</font> being   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c3">K</font><b> such that </b><br/><a NAME="E5:64"/><i><font color="Green" title="E55">A3</font></i>: 
( <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> )
 <b>and </b><br/><a NAME="E6:64"/><i><font color="Green" title="E56">A4</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 <b>and </b><br/><a NAME="E7:64"/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E3:64"><i><font color="Green" title="E54">A2</font></i></a></i>;<br/>
<a NAME="E8:64"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="matrix15.html#E1:64"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<b>hence </b> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix15.html#E6:64"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="matrix15.html#E5:64"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="matrix15.html#E1:64"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>
;<br/><br/>


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