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<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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/>

<b>assume </b><b>that </b><br/><a NAME="E1:48_4_2"/><i><font color="Green">E28</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> 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#K5">-</a> <span class="p1">(<span class="default"><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></span>)</span>
 <b>and </b><br/><a NAME="E2:48_4_2"/><i><font color="Green">E29</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> holds <br/><font color="Maroon">c<sub>5</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#K5">-</a> <span class="p1">(<span class="default"><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></span>)</span>
 ;<br/>

<a NAME="E3:48_4_2"/>
(  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E4:48_4_2"/><b>then </b><i><font color="Green">E30</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> &amp;  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2" target="_self">Indices</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" target="_self">Th25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:48_4_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</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/><a NAME="E2:48_4_2_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E1:48_4_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="matrix_1.html#E2:48_4_2"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_4_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:48_4_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="matrix_1.html#E4:48_4_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="matrix_1.html#T28" target="_self">Th28</a></i>;<br/>


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