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<b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM6">Matrix</a> of <font color="Maroon">m</font>,1,<font color="Maroon">L</font>, <font color="Maroon">M2</font> be   <a href="matrix_1.html#NM6">Matrix</a> of <font color="Maroon">m</font>,1,<font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:44_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E2:44_1_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_2_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_1"><b>suppose </b></a><a NAME="E1:44_1_2_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b><i><font color="Green">E62</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> = 
0
<b>by </b><i><a class="ref" href="matrix_1.html#T23">MATRIX_1:23</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#T23">MATRIX_1:23</a></i>
;<br/><i><font color="Green">E63</font></i>:  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> = 
0
<b>by </b><i>, <a class="ref" href="matrix_1.html#T23">MATRIX_1:23</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#T23">MATRIX_1:23</a></i>
;<br/><a NAME="E4:44_1_2_1_1"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font> holds <br/><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T23">MATRIX_1:23</a></i>;<br/><b>hence </b><a NAME="E5:44_1_2_1_1"/>
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="polynom8.html#T2" target="_self">Th2</a>, <a class="ref" href="polynom8.html#T3" target="_self">Th3</a>, <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_2"><b>suppose </b></a><a NAME="E1:44_1_2_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then </b><i><font color="Green">E65</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> = 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>
;<br/><i><font color="Green">E66</font></i>:  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> = 
1
<b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:44_1_2_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 ;<br/><a NAME="E2:44_1_2_1_2_3"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/><a NAME="E3:44_1_2_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E4:44_1_2_1_2_3"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E5:44_1_2_1_2_3"/><b>then </b><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:44_1_2_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E7:44_1_2_1_2_3"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><a NAME="E8:44_1_2_1_2_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:44_1_2_1_2_3"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><b>hence </b><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="polynom8.html#T4" target="_self">Th4</a>, </i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:44_1_2_1_2"/>
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="polynom8.html#T2" target="_self">Th2</a>, <a class="ref" href="polynom8.html#T3" target="_self">Th3</a>, <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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