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<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>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:48_3_2"/><i><font color="Green">E28</font></i>: 
 <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T27" target="_self">Th27</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:48_3_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:48_3_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:48_3_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:48_3_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><i><font color="Green">E33</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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font>
 ;<br/><div><i><font color="Green">E35</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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:48_3_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E1:48_3_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="matrix_1.html#E2:48_3_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="matrix_1.html#E4:48_3_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="matrix_1.html#E1:48_3_2_1"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_3_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:48_3_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E1:48_3_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="matrix_1.html#E3:48_3_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrix_1.html#E5:48_3_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="matrix_1.html#E1:48_3_2_1"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_3_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:48_3_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E2:48_3_2_1"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:48_3_2"/>
(  <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>2</sub></font> &amp;  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</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>2</sub></font> &amp;  <a href="matrix_1.html#K1" target="_self">width</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E8:48_3_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E6:48_3_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="matrix_1.html#T21" target="_self">Th21</a></i>;<br/>


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