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<b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>, <font color="Maroon">M2</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/>

<a NAME="E1:48_3_2"/><i><font color="Green">E61</font></i>: 
 <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">M2</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:48_3_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">i</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">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">M1</font> holds <br/><font color="Maroon">M1</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">i</font>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E3:48_3_2"/><i><font color="Green">E63</font></i>: 
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" target="_self">Indices</a> <font color="Maroon">M1</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> holds <br/><font color="Maroon">M1</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E4:48_3_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">i</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">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">M2</font> holds <br/><font color="Maroon">M2</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">i</font>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E5:48_3_2"/><i><font color="Green">E65</font></i>: 
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" target="_self">Indices</a> <font color="Maroon">M2</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> holds <br/><font color="Maroon">M2</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 ;<br/>

<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:48_3_2_1"/><i><font color="Green">E84</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>  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">M1</font>
 ;<br/><div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:48_3_2_1_1"/><b>then </b>
( <font color="Maroon">M1</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font> &amp; <font color="Maroon">M2</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">K</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E3:48_3_2_1_1"/>
<font color="Maroon">M1</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:48_3_2_1_2"/><b>then </b>
( <font color="Maroon">M1</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> &amp; <font color="Maroon">M2</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E3:48_3_2_1_2"/>
<font color="Maroon">M1</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:48_3_2_1"/>
<font color="Maroon">M1</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:48_3_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:48_3_2"/>
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 <b>by </b><i>, </i>;<br/>


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