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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:26"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="matrix_8.html#V2" target="_self">Nilpotent</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="matrix_8.html#V2" target="_self">Nilpotent</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_6.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K11">0.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K11">0.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_8.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:26"/><i><font color="Green">E8</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green">E9</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green">E10</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E6:26"/><i><font color="Green">E11</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_6.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_6.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E7:26"/><i><font color="Green">E12</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E1:26"><i><font color="Green">E6</font></i></a>, <a class="txt" href="matrix_8.html#E3:26"><i><font color="Green">E8</font></i></a>, <a class="txt" href="matrix_8.html#E4:26"><i><font color="Green">E9</font></i></a>, <a class="ref" href="matrix_4.html#T62">MATRIX_4:62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E1:26"><i><font color="Green">E6</font></i></a>, <a class="txt" href="matrix_8.html#E3:26"><i><font color="Green">E8</font></i></a>, <a class="ref" href="matrix_4.html#T63">MATRIX_4:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E1:26"><i><font color="Green">E6</font></i></a>, <a class="txt" href="matrix_8.html#E3:26"><i><font color="Green">E8</font></i></a>, <a class="ref" href="matrix_4.html#T63">MATRIX_4:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E5:26"><i><font color="Green">E10</font></i></a>, <a class="txt" href="matrix_8.html#E6:26"><i><font color="Green">E11</font></i></a>, <a class="txt" href="matrix_8.html#E7:26"><i><font color="Green">E12</font></i></a>, <a class="ref" href="matrix_3.html#T5">MATRIX_3:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p3">(<span class="default"><a href="matrix_6.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E1:26"><i><font color="Green">E6</font></i></a>, <a class="txt" href="matrix_8.html#E5:26"><i><font color="Green">E10</font></i></a>, <a class="txt" href="matrix_8.html#E6:26"><i><font color="Green">E11</font></i></a>, <a class="ref" href="matrix_3.html#T5">MATRIX_3:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E1:26"><i><font color="Green">E6</font></i></a>, <a class="txt" href="matrix_8.html#E6:26"><i><font color="Green">E11</font></i></a>, <a class="ref" href="matrix_4.html#T2">MATRIX_4:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K11">0.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_6.html#K2">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K11">0.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E2:26"><i><font color="Green">E7</font></i></a>, <a class="ref" href="matrix_3.html#T6">MATRIX_3:6</a></i>
<br/>.= 
 <a href="matrix_3.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="matrix_3.html#T6">MATRIX_3:6</a></i>
<br/>.= 
 <a href="matrix_1.html#K11">0.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>

;<br/>
<b>hence </b><a NAME="E9:26"/>
<font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> is <a href="matrix_8.html#V2" target="_self">Nilpotent</a>
 <b>by </b><i><a class="ref" href="matrix_8.html#D2" target="_self">Def2</a></i>;<br/>


</div>
